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Volume Problem Solving

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To solve problems on this page, you should be familiar with the following: Volume - Cuboid Volume - Sphere Volume - Cylinder Volume - Pyramid

This wiki includes several problems motivated to enhance problem-solving skills. Before getting started, recall the following formulas:

• Volume of sphere with radius \(r:\) \( \frac43 \pi r^3 \)
• Volume of cube with side length \(L:\) \( L^3 \)
• Volume of cone with radius \(r\) and height \(h:\) \( \frac13\pi r^2h \)
• Volume of cylinder with radius \(r\) and height \(h:\) \( \pi r^2h\)
• Volume of a cuboid with length \(l\), breadth \(b\), and height \(h:\) \(lbh\)

Volume Problem Solving - Basic

Volume - problem solving - intermediate, volume problem solving - advanced.

This section revolves around the basic understanding of volume and using the formulas for finding the volume. A couple of examples are followed by several problems to try.

Find the volume of a cube of side length \(10\text{ cm}\). \[\begin{align} (\text {Volume of a cube}) & = {(\text {Side length}})^{3}\\ & = {10}^{3}\\ & = 1000 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

Find the volume of a cuboid of length \(10\text{ cm}\), breadth \(8\text{ cm}\). and height \(6\text{ cm}\). \[\begin{align} (\text {Area of a cuboid}) & = l × b × h\\ & = 10 × 8 × 6\\ & = 480 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

I made a large ice cream cone of a composite shape of a cone and a hemisphere. If the height of the cone is 10 and the diameter of both the cone and the hemisphere is 6, what is the volume of this ice cream cone? The volume of the composite figure is the sum of the volume of the cone and the volume of the hemisphere. Recall the formulas for the following two volumes: \( V_{\text{cone}} = \frac13 \pi r^2 h\) and \( V_{\text{sphere}} =\frac43 \pi r^3 \). Since the volume of a hemisphere is half the volume of a a sphere of the same radius, the total volume for this problem is \[\frac13 \pi r^2 h + \frac12 \cdot \frac43 \pi r^3. \] With height \(h =10\), and diameter \(d = 6\) or radius \(r = \frac d2 = 3 \), the total volume is \(48\pi. \ _\square \)

Find the volume of a cone having slant height \(17\text{ cm}\) and radius of the base \(15\text{ cm}\). Let \(h\) denote the height of the cone, then \[\begin{align} (\text{slant height}) &=\sqrt {h^2 + r^2}\\ 17&= \sqrt {h^2 + 15^2}\\ 289&= h^2 + 225\\ h^2&=64\\ h& = 8. \end{align}\] Since the formula for the volume of a cone is \(\dfrac {1}{3} ×\pi ×r^2×h\), the volume of the cone is \[ \frac {1}{3}×3.14× 225 × 8= 1884 ~\big(\text{cm}^{2}\big). \ _\square\]

Find the volume of the following figure which depicts a cone and an hemisphere, up to \(2\) decimal places. In this figure, the shape of the base of the cone is circular and the whole flat part of the hemisphere exactly coincides with the base of the cone (in other words, the base of the cone and the flat part of the hemisphere are the same). Use \(\pi=\frac{22}{7}.\) \[\begin{align} (\text{Volume of cone}) & = \dfrac {1}{3} \pi r^2 h\\ & = \dfrac {1 × 22 × 36 × 8}{3 × 7}\\ & = \dfrac {6336}{21} = 301.71 \\\\ (\text{Volume of hemisphere}) & = \dfrac {2}{3} \pi r^3\\ & = \dfrac {2 × 22 × 216}{3 × 7}\\ & = \dfrac {9504}{21} = 452.57 \\\\ (\text{Total volume of figure}) & = (301.71 + 452.57) \\ & = 754.28.\ _\square \end{align} \]

Try the following problems.

Find the volume (in \(\text{cm}^3\)) of a cube of side length \(5\text{ cm} \).

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

Bob has a pipe with a diameter of \(\frac { 6 }{ \sqrt { \pi } }\text{ cm} \) and a length of \(3\text{ m}\). How much water could be in this pipe at any one time, in \(\text{cm}^3?\)

What is the volume of the octahedron inside this \(8 \text{ in}^3\) cube?

A sector with radius \(10\text{ cm}\) and central angle \(45^\circ\) is to be made into a right circular cone. Find the volume of the cone.

\[\] Details and Assumptions:

• The arc length of the sector is equal to the circumference of the base of the cone.

Three identical tanks are shown above. The spheres in a given tank are the same size and packed wall-to-wall. If the tanks are filled to the top with water, then which tank would contain the most water?

A chocolate shop sells its products in 3 different shapes: a cylindrical bar, a spherical ball, and a cone. These 3 shapes are of the same height and radius, as shown in the picture. Which of these choices would give you the most chocolate?

\[\text{ I. A full cylindrical bar } \hspace{.4cm} \text{ or } \hspace{.45cm} \text{ II. A ball plus a cone }\]

How many cubes measuring 2 units on one side must be added to a cube measuring 8 units on one side to form a cube measuring 12 units on one side?

This section involves a deeper understanding of volume and the formulas to find the volume. Here are a couple of worked out examples followed by several "Try It Yourself" problems:

\(12\) spheres of the same size are made from melting a solid cylinder of \(16\text{ cm}\) diameter and \(2\text{ cm}\) height. Find the diameter of each sphere. Use \(\pi=\frac{22}{7}.\) The volume of the cylinder is \[\pi× r^2 × h = \frac {22×8^2×2}{7}= \frac {2816}{7}.\] Let the radius of each sphere be \(r\text{ cm}.\) Then the volume of each sphere in \(\text{cm}^3\) is \[\dfrac {4×22×r^3}{3×7} = \dfrac{88×r^3}{21}.\] Since the number of spheres is \(\frac {\text{Volume of cylinder}}{\text {Volume of 1 sphere}},\) \[\begin{align} 12 &= \dfrac{2816×21}{7×88×r^3}\\ &= \dfrac {96}{r^3}\\ r^3 &= \dfrac {96}{12}\\ &= 8\\ \Rightarrow r &= 2. \end{align}\] Therefore, the diameter of each sphere is \[2\times r = 2\times 2 = 4 ~(\text{cm}). \ _\square\]

Find the volume of a hemispherical shell whose outer radius is \(7\text{ cm}\) and inner radius is \(3\text{ cm}\), up to \(2\) decimal places. We have \[\begin{align} (\text {Volume of inner hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × R^3\\ & = \dfrac {1 × 4 × 22 × 27}{2 × 3 × 7}\\ & = \dfrac {396}{7}\\ & = 56.57 ~\big(\text{cm}^{3}\big) \\\\ (\text {Volume of outer hemisphere}) & = \dfrac{1}{2} × \dfrac{4}{3} × \pi × r^3\\ & = \dfrac {1 × 4 × 22 × 343}{2 × 3 × 7}\\ & = \dfrac {2156}{7}\\ & = 718.66 ~\big(\text{cm}^{3}\big) \\\\ (\text{Volume of hemispherical shell}) & = (\text{V. of outer hemisphere}) - (\text{V. of inner hemisphere})\\ & = 718.66 - 56.57 \\ & = 662.09 ~\big(\text{cm}^{3}\big).\ _\square \end{align}\]

A student did an experiment using a cone, a sphere, and a cylinder each having the same radius and height. He started with the cylinder full of liquid and then poured it into the cone until the cone was full. Then, he began pouring the remaining liquid from the cylinder into the sphere. What was the result which he observed?

There are two identical right circular cones each of height \(2\text{ cm}.\) They are placed vertically, with their apex pointing downwards, and one cone is vertically above the other. At the start, the upper cone is full of water and the lower cone is empty.

Water drips down through a hole in the apex of the upper cone into the lower cone. When the height of water in the upper cone is \(1\text{ cm},\) what is the height of water in the lower cone (in \(\text{cm}\))?

On each face of a cuboid, the sum of its perimeter and its area is written. The numbers recorded this way are 16, 24, and 31, each written on a pair of opposite sides of the cuboid. The volume of the cuboid lies between \(\text{__________}.\)

A cube rests inside a sphere such that each vertex touches the sphere. The radius of the sphere is \(6 \text{ cm}.\) Determine the volume of the cube.

If the volume of the cube can be expressed in the form of \(a\sqrt{3} \text{ cm}^{3}\), find the value of \(a\).

A sphere has volume \(x \text{ m}^3 \) and surface area \(x \text{ m}^2 \). Keeping its diameter as body diagonal, a cube is made which has volume \(a \text{ m}^3 \) and surface area \(b \text{ m}^2 \). What is the ratio \(a:b?\)

Consider a glass in the shape of an inverted truncated right cone (i.e. frustrum). The radius of the base is 4, the radius of the top is 9, and the height is 7. There is enough water in the glass such that when it is tilted the water reaches from the tip of the base to the edge of the top. The proportion of the water in the cup as a ratio of the cup's volume can be expressed as the fraction \( \frac{m}{n} \), for relatively prime integers \(m\) and \(n\). Compute \(m+n\).

The square-based pyramid A is inscribed within a cube while the tetrahedral pyramid B has its sides equal to the square's diagonal (red) as shown.

Which pyramid has more volume?

Please remember this section contains highly advanced problems of volume. Here it goes:

Cube \(ABCDEFGH\), labeled as shown above, has edge length \(1\) and is cut by a plane passing through vertex \(D\) and the midpoints \(M\) and \(N\) of \(\overline{AB}\) and \(\overline{CG}\) respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are relatively prime positive integers. Find \(p+q\).

If the American NFL regulation football

has a tip-to-tip length of \(11\) inches and a largest round circumference of \(22\) in the middle, then the volume of the American football is \(\text{____________}.\)

Note: The American NFL regulation football is not an ellipsoid. The long cross-section consists of two circular arcs meeting at the tips. Don't use the volume formula for an ellipsoid.

Answer is in cubic inches.

Consider a solid formed by the intersection of three orthogonal cylinders, each of diameter \( D = 10 \).

What is the volume of this solid?

Consider a tetrahedron with side lengths \(2, 3, 3, 4, 5, 5\). The largest possible volume of this tetrahedron has the form \( \frac {a \sqrt{b}}{c}\), where \(b\) is an integer that's not divisible by the square of any prime, \(a\) and \(c\) are positive, coprime integers. What is the value of \(a+b+c\)?

Let there be a solid characterized by the equation \[{ \left( \frac { x }{ a } \right) }^{ 2.5 }+{ \left( \frac { y }{ b } \right) }^{ 2.5 } + { \left( \frac { z }{ c } \right) }^{ 2.5 }<1.\]

Calculate the volume of this solid if \(a = b =2\) and \(c = 3\).

• Surface Area

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Volume Word Problems

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• Maths Questions

Volume Questions

Volume questions and answers are available in an easily understandable format along with required formulas. Students can practise questions on finding the volume of various solids provided here, and verify their solutions with the available answers. This is the best way to learn about various problem-solving techniques of solids in geometry.

What is volume?

Volume is the amount of space occupied by a solid shape in a three-dimensional plane or region. Some examples of solids include cubes, cuboids, spheres, cones, cylinders, etc.

• Volume of cube = a 3 , where a is the edge of the cube.
• Volume of cuboid = lbh, where l = length, b = breadth and h = height.
• Volume of sphere = (4/3) πr 3 , where r is the radius of the sphere.
• Volume of cylinder = πr 2 h, where r is the radius of the circular bases and h is the height.
• Volume of cone = (⅓) πr 2 h, where r is the radius of the circular base and h is the height.
• Volume of hemisphere = (2/3) πr 3 , where r is the radius of the sphere.
• Volume of frustum = πh/3 (R 2 + r 2 + Rr), where ‘R’ and ‘r’ are the radii of the base and top of the frustum.
• Volume of prism = Base Area x Height.
• Volume of pyramid = ⅓ (Area of base) (Height).

Click here to learn more about volume .

Volume Questions and Answers

1. Find the volume of a cuboidal box with dimensions 11 cm × 8 cm × 13 cm.

Given, the dimensions of a cuboidal box: 11 cm × 8 cm × 13 cm

Length = l = 11 cm

Breadth = b = 8 cm

Height = h = 13 cm

As we know, volume of a cuboid = lbh

= 11 × 8 × 13

Thus, the volume of the cuboidal box is 1144 cm 3 .

2. What is the volume of a sphere of diameter 21 units?

Diameter of the sphere = 21 units

Radius of the sphere = (21/2) units

Volume of a sphere = (4/3) πr 3

= (4/3) × (22/7) × (21/2) × (21/2) × (21/2)

Therefore, the volume of the sphere is 4851 cubic units.

3. A cuboidal block of wood was cut into eight equal cubes of edges 4 cm. Find the volume of the initial block of wood.

Edge of a cubical wood = 4 cm

Volume of the cube = a 3

= 4 × 4 × 4

Volume of 8 such cubes = 8 volume of one cube

Hence, the volume of the initial cuboidal block of wood is 512 cm 3 .

4. If three solid spherical beads of radii 3 cm, 4 cm, and 5 cm, respectively, are melted into one spherical bead, then find its radius in cm.

Radii of three solid spherical beads: 3 cm, 4 cm and 5 cm

As we know, the volume of a sphere = (4/3) πr 3

Let R be the radius of the new spherical bead, which is made by melting three spherical beads.

Volume of a new spherical bead = Sum of volumes of three spherical beads

(4/3)πR 3 = (4/3) × 3 × 3 × 3 + (4/3) × 4 × 4 × 4 + (4/3) × 5 × 5 × 5

(4/3)πR 3 = (4/3)π (27 + 64 + 125)

Hence, the radius of the spherical bead is 6 cm.

5. How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to build a wall of 8 m x 6 m x 22.5 cm?

Dimensions of a brick = 25 cm x 11.25 cm x 6 cm

Dimensions of a wall = 8 m x 6 m x 22.5 cm

= 800 cm x 600 cm x 22.5 cm

Number of bricks = volume of the wall/ volume of a brick

= (800 x 600 x 22.5) / (25 x 11.25 x 6)

6. Find the depth of the cylindrical tank if its capacity is 1848 m 3 and the diameter of the base is 14 m.

Let h be the depth of the cylindrical tank.

Diameter of the base = 14 m

Radius of the circular base = r = 14/2 = 7 m

Volume of the cylindrical tank = πr2h

So, πr 2 h = 1848 m 3

(22/7) × 7 × 7 × h = 1848

h = 1848/(22 × 7)

Therefore, the depth of the cylindrical tank is 12 m.

7. What is the ratio of the volume of a cone, a sphere and a cylinder if each has the same radius and height?

Let r be the radius of the cone, sphere and cylinder.

Let h be the height of the cone and cylinder.

Also, r = h

We know that,

Volume of cone = (1/3)πr 2 h = (1/3)πr 3

Volume of sphere = (4/3)πr 3

Volume of cylinder = πr 2 h = πr 3

Volume of cone : Volume of sphere : Volume of cylinder

= (1/3)πr 2 h : (4/3)πr 3 : πr 2 h

= (1/3)πr 3 : (4/3)πr 3 : πr 3

= (1/3) : (4/3) : 1

= 1 : 4 : 3

Therefore, the required ratio is 1 : 4 : 3.

8. Find the water weight in a conical vessel that is 21 cm deep and 16 cm in diameter.

Depth of height of a conical vessel = h = 21 cm

Diameter of the circular base = 16 cm

Radius = r = 16/2 = 8 cm

Volume of a cone = (1/3)πr 2 h

= (1/3) × (22/7) × 8 × 8 × 21

= 1408 cm 3

= (1408/1000) kg

Therefore, the weight of the water in a conical vessel is 1.408 kg.

9. A vessel is in the form of a frustum of a cone. Its radius at one end and the heights are 8 cm and 14 cm, respectively. If its volume is 5676/3 cm 3 , find the radius at the other end.

Radius of one circular end = r 1 = 8 cm

Height of the frustum = h = 14 cm

Let r2 be the radius of the other circular end.

Volume of the frustum = πh/3 (r 1 2 + r 2 2 + r 1 r 2 )

= (1/3) × (22/7) × 14 × [(8) 2 + (r 2 ) 2 + 8r 2 ]

According to the given,

(1/3) × (22/7) × 14 × [(8) 2 + (r 2 ) 2 + 8r 2 ] = 5676/3 cm3

64 + r 2 2 + 8r 2 = 5676/44

r 2 2 + 8r 2 = 129 – 64

r 2 2 + 8r 2 – 65 = 0

r 2 2 + 13r 2 – 5r 2 – 65 = 0

r 2 (r 2 + 13) – 5(r 2 + 13) = 0

(r 2 – 5)(r 2 + 13) = 0

r 2 = 5, r 2 = -13

Hence, the radius of the other circular end is 5 cm.

10. If the curved surface area of a right circular cone is 10010 cm 2 and its radius is 35 cm, find the volume of the cone.

Radius of the circular base of a cone = r = 35 cm

Let h and l be the height and slant height of the right circular cone.

Also, given that the curved surface area = 10010 cm 2

πrl = 10010

(22/7) × 35 × l = 10010

l = (10010 × 7)/ (22 × 35)

As we know, l 2 = r 2 + h 2

h 2 = l 2 – r 2

= (91) 2 – (35) 2

= 8281 – 1225

Volume of the cone = (1/3) πr 2 h

= (1/3) × (22/7) × 35 × 35 × 84

Hence, the volume of the right circular cone is 107800 cm 3 .

Practice Questions on Volume

• The height of the wall is six times its width, and the length of the wall is seven times its height. If the volume of the wall is 16128 cm 3 , what is the width of the wall?
• A fish tank has a length of 45 cm, a width of 25 cm, and a depth of 10 cm. Calculate the volume of the fish tank.
• Find the volume of the cylinder with a radius of 5.4 units and a height of 16 units.
• Find the number of small cubes with edges of 10 cm that can be accommodated in a cubical box with a 1 m edge.
• The radius of a sphere is increased by 10%. Prove that the volume will be increased by 33.1% approximately.

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5th Grade Volume Worksheets

Welcome to our 5th Grade Volume Worksheets page.

Here you will find our collection of worksheets to introduce and help you learn about volume.

These worksheets will help you to understand and practice how to find the volume of rectangular prisms and other simple shapes.

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Volume of Rectangular Prisms

On this webpage you will find our range of worksheets to help you work out the volume of simple 3d shapes such as rectangular prisms.

They are also very useful for introducing the concept of volume being the number of cubes that fill up a space.

These sheets are graded from easiest to hardest, and each sheet comes complete with answers.

Using these sheets will help your child to:

• learn how to find the volume of simple 3d shapes by counting cubes;
• learn how to find the volume of rectangular prisms by multiplying length x width x height
• practice using their knowledge to solve basic volume problems.

What is Volume?

• Volume is the amount of space that is inside a 3 dimensional shape.
• Because it is an amount of space, it has to be measured in cubes.
• If the shape is measured in cm, then the volume would be measured in cubic cm or cm 3
• If the shape is measured in inches, then the volume would be measured in cubic inches or in 3

Volume of a Rectangular Prism

• The volume of a rectangular prism is the number of cubes it is made from.
• To find the number of cubes, we need to multiply the length by the width by the height.
• So Volume = length x width x height or l x w x h.
• We could also multiply the area of the base (which is the length x width) by the height.
• So Volume = l x w x h or b x h (where b is the area of the base)

In the example above, the length is 3, the width is 6 and the height is 2.

So the volume is 3 x 6 x 2 = 36cm 3 or 36 cubic cm.

This tells us that there are 36 cm cubes that make up the shape.

We have split our worksheets up into different sections, to make it easier for you to select the right sheets for your needs.

• Section 1 - Find the Volume by Counting Cubes
• Section 2 - Finding the Volume by multiplication
• Section 3 - Match the Volume (multiplication)
• Section 3 - Volume Problem Solving Riddles

5th Grade Volume Worksheets - Counting Cubes

• Volume - Count the Cubes Sheet 1
• PDF version
• Volume - Count the Cubes Sheet 2

5th Grade Volume Worksheets - Find the Volume by Multiplication

The first sheet is supported, the other two sheets are more independent.

You can choose between the standard or metric versions of sheets 2 and 3 (the measurements are the same)

• Find the Volume Sheet 1 (supported)
• Find the Volume Sheet 2 (standard)
• Find the Volume Sheet 2 (metric)
• Find the Volume Sheet 3 (standard)
• Find the Volume Sheet 3 (metric)

5th Grade Volume Worksheets - Match the Volume

• Match the Volume Sheet 1
• Match the Volume Sheet 2

5th Grade Volume Worksheets - Volume Riddles

• Volume Riddles Sheet 5A
• Volume Riddles Sheet 5B

Volume of Rectangular Prisms Walkthrough Video

This short video walkthrough shows several problems from our Find the Volume Sheet 2 being solved and has been produced by the West Explains Best math channel.

If you would like some support in solving the problems on these sheets, check out the video below!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Volume of a Cube/Cuboid/Box Calculators

Each of the pages below includes a handy calculator to help you find the volume of cubes, cuboids and boxes.

• Volume of a Cube Calculator

• Volume of a Box Calculator

Converting Measures Worksheets

Here is our selection of converting units of measure for 3rd to 5th graders.

These sheets involve converting between customary units of measure and also metric units.

• Converting Customary Units Worksheets
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5th Grade Geometry Worksheets

Here is our selection of 5th grade Geometry worksheets about angles.

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• 5th Grade Geometry Missing Angles

Area Worksheets

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• work out the areas of a range of rectangles;
• find the area of rectilinear shapes.
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Here is our selection of free printable perimeter worksheets for 3rd and 4th grade.

• work out the perimeter of a range of rectangles;
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All the math practice worksheets in this section support Elementary Math Benchmarks.

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Volume Worksheets

This humongous collection of printable volume worksheets is sure to walk middle and high school students step-by-step through a variety of exercises beginning with counting cubes, moving on to finding the volume of solid shapes such as cubes, cones, rectangular and triangular prisms and pyramids, cylinders, spheres and hemispheres, L-blocks, and mixed shapes. Brimming with learning and backed by application the PDFs offer varied levels of difficulty.

List of Volume Worksheets

Counting Cubes

• Volume of Cubes
• Volume of Rectangular Prisms
• Volume of Triangular Prisms

Volume of Mixed Prisms

• Volume of Cones
• Volume of Cylinders

Volume of Spheres and Hemispheres

• Volume of Rectangular Pyramids
• Volume of Triangular Pyramids

Volume of Mixed Pyramids

Volume of Mixed Shapes

Volume of Composite Shapes

Explore the Volume Worksheets in Detail

Work on the skill of finding volume with this batch of counting cubes worksheets. Count unit cubes to determine the volume of rectangular prisms and solid blocks, draw prisms on isometric dot paper and much more.

Volume of a Cube

Augment practice with this unit of pdf worksheets on finding the volume of a cube comprising problems presented as shapes and in the word format with side length measures involving integers, decimals and fractions.

Volume of a Rectangular Prism

This batch of volume worksheets provides a great way to learn and perfect skills in finding the volume of rectangular prisms with dimensions expressed in varied forms, find the volume of L-blocks, missing measure and more.

Volume of a Triangular Prism

Encourage students to work out the entire collection of printable worksheets on computing the volume of triangular prism using the area of the cross-section or the base and leg measures and practice unit conversions too.

Navigate through this collection of volume of mixed prism worksheets featuring triangular, rectangular, trapezoidal and polygonal prisms. Bolster practice with easy and moderate levels classified based on the number range used.

Volume of a Cone

Motivate learners to use the volume of a cone formula efficiently in the easy level, find the radius in the moderate level and convert units in the difficult level, solve for volume using slant height, and find the volume of a conical frustum too.

Volume of a Cylinder

Access our volume of a cylinder worksheets to practice finding the radius from diameter, finding the volume of cylinders with parameters in integers and decimals, find the missing parameters, solve word problems and more!

Take the hassle out of finding the volume of spheres and hemispheres with this compilation of pdf worksheets. Gain immense practice with a wide range of exercises involving integers and decimals.

Volume of a Rectangular Pyramid

This exercise is bound to help learners work on the skill of finding the volume of rectangular pyramids with dimensions expressed as integers, decimals and fractions in easy and moderate levels.

Volume of a Triangular Pyramid

Help children further their practice with this bundle of pdf worksheets on determining the volume of triangular pyramids using the measures of the base area or height and base. The problems are offered as 3D shapes and in word format in varied levels of difficulty.

Gain ample practice in finding the volume of pyramids with triangular, rectangular and polygonal base faces presented in two levels of difficulty. Apply relevant formulas to find the volume using the base area or the other dimensions provided.

Upscale practice with an enormous collection of printable worksheets on finding the volume of solid shapes like prisms, cylinders, cones, pyramids and revision exercises to revisit concepts with ease.

Learn to find the volume of composite shapes that are a combination of two or more solid 3D shapes. Begin with counting squares, find the volume of L -blocks, and compound shapes by adding or subtracting volumes of decomposed shapes.

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9.9: Solve Geometry Applications- Volume and Surface Area (Part 1)

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• Page ID 5010

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Learning Objectives

• Find volume and surface area of rectangular solids
• Find volume and surface area of spheres
• Find volume and surface area of cylinders
• Find volume of cones

be prepared!

Before you get started, take this readiness quiz.

• Evaluate x 3 when x = 5. If you missed this problem, review Example 2.3.3 .
• Evaluate 2 x when x = 5. If you missed this problem, review Example 2.3.4 .
• Find the area of a circle with radius \(\dfrac{7}{2}\). If you missed this problem, review Example 5.6.12 .

In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

Problem Solving Strategy for Geometry Applications

• Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
• Step 2. Identify what you are looking for.
• Step 3. Name what you are looking for. Choose a variable to represent that quantity.
• Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer in the problem and make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See Figure \(\PageIndex{1}\)). The amount of paint needed to cover the outside of each box is the surface area , a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

Figure \(\PageIndex{1}\) - This wooden crate is in the shape of a rectangular solid.

Each crate is in the shape of a rectangular solid . Its dimensions are the length, width, and height. The rectangular solid shown in Figure \(\PageIndex{2}\) has length 4 units, width 2 units, and height 3 units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Figure \(\PageIndex{2}\) - Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This 4 by 2 by 3 rectangular solid has 24 cubic units.

Altogether there are 24 cubic units. Notice that 24 is the length × width × height.

The volume, V, of any rectangular solid is the product of the length, width, and height.

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, B, is equal to length × width.

\[B = L \cdot W\]

We can substitute B for L • W in the volume formula to get another form of the volume formula.

\[\begin{split} V &= \textcolor{red}{L \cdot W} \cdot H \\ V &= \textcolor{red}{(L \cdot W)} \cdot H \\ V &= \textcolor{red}{B} h \end{split}\]

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the 4 × 2 × 3 rectangular solid we started with. See Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\)

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

\[\begin{split} A_{front} &= L \times W \qquad A_{side} = L \times W \qquad A_{top} = L \times W \\ A_{front} &= 4 \cdot 3 \qquad \quad \; A_{side} = 2 \cdot 3 \qquad \quad \; A_{top} = 4 \cdot 2 \\ A_{front} &= 12 \qquad \qquad A_{side} = 6 \qquad \qquad \; \; A_{top} = 8 \end{split}\]

Notice for each of the three faces you see, there is an identical opposite face that does not show.

\[\begin{split} S &= (front + back)+(left\; side + right\; side) + (top + bottom) \\ S &= (2 \cdot front) + (2 \cdot left\; side) + (2 \cdot top) \\ S &= 2 \cdot 12 + 2 \cdot 6 + 2 \cdot 8 \\ S &= 24 + 12 + 16 \\ S &= 52\; sq.\; units \end{split}\]

The surface area S of the rectangular solid shown in Figure \(\PageIndex{3}\) is 52 square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see Figure \(\PageIndex{4}\)). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

\[S = 2LH + 2LW + 2WH\]

Figure \(\PageIndex{4}\) - For each face of the rectangular solid facing you, there is another face on the opposite side. There are 6 faces in all.

Definition: Volume and Surface Area of a Rectangular Solid

For a rectangular solid with length L, width W, and height H:

Example \(\PageIndex{1}\):

For a rectangular solid with length 14 cm, height 17 cm, and width 9 cm, find the (a) volume and (b) surface area.

Step 1 is the same for both (a) and (b), so we will show it just once.

Exercise \(\PageIndex{1}\):

Find the (a) volume and (b) surface area of rectangular solid with the: length 8 feet, width 9 feet, and height 11 feet.

Exercise \(\PageIndex{2}\):

Find the (a) volume and (b) surface area of rectangular solid with the: length 15 feet, width 12 feet, and height 8 feet.

1,440 cu. ft

Example \(\PageIndex{2}\):

A rectangular crate has a length of 30 inches, width of 25 inches, and height of 20 inches. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{3}\):

A rectangular box has length 9 feet, width 4 feet, and height 6 feet. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{4}\):

A rectangular suitcase has length 22 inches, width 14 inches, and height 9 inches. Find its (a) volume and (b) surface area.

2,772 cu. in

1,264 sq. in.

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

\[\begin{split} V &= LWH \quad \; S = 2LH + 2LW + 2WH \\ V &= s \cdot s \cdot s \quad S = 2s \cdot s + 2s \cdot s + 2s \cdot s \\ V &= s^{3} \qquad \quad S = 2s^{2} + 2s^{2} + 2s^{2} \\ &\qquad \qquad \quad \; S = 6s^{2} \end{split}\]

So for a cube, the formulas for volume and surface area are V = s 3 and S = 6s 2 .

Definition: Volume and Surface Area of a Cube

For any cube with sides of length s,

Example \(\PageIndex{3}\):

A cube is 2.5 inches on each side. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{5}\):

For a cube with side 4.5 meters, find the (a) volume and (b) surface area of the cube.

91.125 cu. m

121.5 sq. m

Exercise \(\PageIndex{6}\):

For a cube with side 7.3 yards, find the (a) volume and (b) surface area of the cube.

389.017 cu. yd.

319.74 sq. yd.

Example \(\PageIndex{4}\):

A notepad cube measures 2 inches on each side. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{7}\):

A packing box is a cube measuring 4 feet on each side. Find its (a) volume and (b) surface area.

Exercise \(\PageIndex{8}\):

A wall is made up of cube-shaped bricks. Each cube is 16 inches on each side. Find the (a) volume and (b) surface area of each cube.

4,096 cu. in.

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

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Resources tagged with: Volume and capacity

There are 54 NRICH Mathematical resources connected to Volume and capacity , you may find related items under Measuring and calculating with units .

Pouring Problem

What do you think is going to happen in this video clip? Are you surprised?

Colourful Cube

A colourful cube is made from little red and yellow cubes. But can you work out how many of each?

Compare the Cups

You'll need a collection of cups for this activity.

Bottles (2)

In this activity focusing on capacity, you will need a collection of different jars and bottles.

Bottles (1)

For this activity which explores capacity, you will need to collect some bottles and jars.

Changing Areas, Changing Volumes

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Place Your Orders

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Fill Me up Too

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Discuss and Choose

This activity challenges you to decide on the 'best' number to use in each statement. You may need to do some estimating, some calculating and some research.

Can you sketch graphs to show how the height of water changes in different containers as they are filled?

Can you lay out the pictures of the drinks in the way described by the clue cards?

Next Size Up

The challenge for you is to make a string of six (or more!) graded cubes.

Growing Rectangles

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Maths Filler

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Cuboid Challenge

What's the largest volume of box you can make from a square of paper?

Thousands and Millions

Here's a chance to work with large numbers...

All in a Jumble

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Efficient Cutting

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Sending a Parcel

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Three Cubes

Can you work out the dimensions of the three cubes?

Making Cuboids

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Uniform Units

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

Which of these infinitely deep vessels will eventually full up?

Maths Filler 2

Can you draw the height-time chart as this complicated vessel fills with water?

At the Pumps

How will you find out how much a tank of petrol costs?

Zin Obelisk

In the ancient city of Atlantis a solid rectangular object called a Zin was built in honour of the goddess Tina. Your task is to determine on which day of the week the obelisk was completed.

Cuboid-in-a-box

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

All Wrapped Up

What is the largest cuboid you can wrap in an A3 sheet of paper?

Tubular Stand

If the radius of the tubing used to make this stand is r cm, what is the volume of tubing used?

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Conical Bottle

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

Mouhefanggai

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Volume of a Pyramid and a Cone

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

The Genie in the Jar

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal spoons. Each day a spoonful was used to perfume the bath of a beautiful princess. For how many days did the whole jar last? The genie's master replied: Five hundred and ninety five days. What three numbers do the genie's words granid, ozvik and vaswik stand for?

More Christmas Boxes

What size square should you cut out of each corner of a 10 x 10 grid to make the box that would hold the greatest number of cubes?

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Concrete Calculation

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

Reach for Polydron

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Double Your Popcorn, Double Your Pleasure

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

The Big Cheese

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Multilink Cubes

If you had 36 cubes, what different cuboids could you make?

SOLVING VOLUME PROBLEMS

When we solve a real-world problem involving the volume of a prism, we can  choose to use either of the volume formulas we know.

Example 1 :

A terrarium is shaped like a rectangular prism. The prism is 25  inches long,  13  inches wide, and 16 inches deep. What is the volume of the terrarium ?

From the given information, we can have the following figure. 

Step 2 : 

In the above figure, the base is in the shape of a rectangle with length 25 inches and width 13 inches.  

Therefore, 

Area of the base  =  length x width 

Area of the base  =  25 x 13

Area of the base  =  325 square inches.

Step 3 : 

Volume of the terrarium  =  Volume of rectangular prism 

Volume of the terrarium  =  Base area x Height 

Volume of the terrarium  =  325 x 16

Volume of the terrarium  =  5200 cubic inches

Example 2 :

A rectangular swimming pool is 15 meters long, 10 1/ 2 meters wide,  and 2 1/ 2 meters deep. What is its volume?

In the above figure, the base is in the shape of a rectangle with length 15 meters and width 10 1/2 meters. 

Area of the base  =  15 x 10 1/2

Area of the base  =  15 x 10.5

Area of the base  =  157.5

Area of the base  =  157.5 square inches.

Volume of swimming pool  =  Volume of rectangular prism 

Volume of the  swimming pool   =  Base area x Height 

Volume of the  swimming pool   =  157.5 x 2 1/2

Volume of the  swimming pool   =  157.5 x 2.5

Volume of the swimming pool  =  393.75

Volume of the swimming pool  =  393 75/100

Volume of the swimming pool  =  393 3/4 cubic meters

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GPT-4o responds as fast as humans.

ChatGPT 4 may still be relatively new, but OpenAI is already iterating with an upgrade that can respond as quickly as humans do in normal conversation. The company showed off GPT-4o in a live demo , showing off its ability to use your phone’s camera to solve math equations and deliver a much more conversational voice assistant experience.

While we only have the event demo to go off of, GPT-4o looks impressive. It doesn’t even have to wait for you to finish your request and can roll with interruptions mid-prompt, bringing one step closer to living out Her in real life.

Even Faster Response Times

According to OpenAI, the GPT-4o model can respond as fast as 232 milliseconds to audio inputs. More realistically, it averages around 320 milliseconds to respond, which OpenAI said is similar to how fast humans respond in conversation.

On top of the speed, GPT-4o can handle interruptions and any adjustment requests. As seen in the bedtime story demo, GPT-4o immediately stopped talking when interrupted and quickly handled requests like adding more dramatic inflections, narrating in a robot voice, and even singing the entire prompt out loud. If that demo doesn’t convince you, two GPT-4o models improvising a song together should.

GPT-4o isn’t just more responsive to voice, it can also see better. The new vision features allow it to see through your device’s camera and understand things like handwritten math equations or messages . It’s eerie how genuinely touched GPT-4o sounds when it sees and understands a message that says “I Heart ChatGPT.” Even more impressive, GPT-4o can handle coding tasks and live translations between two people. This should feel way more natural than Google Translate when you’re trying to have a conversation in a foreign country.

Available for Free

OpenAI said the text and image capabilities for GPT-4o roll out today, but the voice feature will be coming to alpha within ChatGPT Plus in the coming weeks. Once it’s fully ready, the upgraded ChatGPT model will be available to all users, subscribers or otherwise. However, if you pay $20 per month for ChatGPT Plus , you’ll get five times the message limits of GPT-4o compared to the free version.

Anytime a large language model gets such an impressive update, we have to consider the potential for misuse . Considering how smoothly the live demo for solving the equation went, it looks like an even better way to help students get out of their math homework. However, OpenAI said that GPT-4o was built with new safety systems to offer guardrails on voice outputs. We’ll have to wait and see if these guardrails are enough.

Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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How to Solve Maths Problems with Google Circle for Students

Math can be a challenging subject for many students. Between understanding word problems and applying the right formulas, getting stuck is a common experience. Google has introduced a powerful tool called Google Circle, designed to help students solve math problems with confidence.

Read In Short: What Google Circle is and how it can help students solve math problems. How to use Google Circle to break down and solve different types of math problems. Additional tips and tricks to maximize your learning experience with Google Circle

What is Google Circle?

Google Circle is a feature within the Google Search app for Android devices . It allows you to solve math problems directly from your phone or tablet. Instead of simply providing answers, Google Circle focuses on math problem-solving by offering step-by-step guidance. This makes it a valuable tool for understanding the concepts behind the calculations, not just getting the final answer.

How Does Google Circle Work

Using Google Circle is incredibly simple. Here’s how it goes:

Step 1: Open the Google Search app on your Android device.

Step 2: find the math problem you’re working on., step 3: circle the specific problem you want help with using your finger., step 4: google circle will recognize it and use its artificial intelligence to analyze it., how to enable circle to search, step 1: open settings on your phone, step 2: scroll down and select display, step 3: scroll down and select navigation bar, step 4: select the toggle and turn on circle to search, which devices are getting the circle to search update.

Below is the list of android which is getting Google Circle App

• Samsung Galaxy S24, S24 Plus, and S24 Ultra
• Samsung Galaxy S23, S23 Plus, and S23 Ultra
• Samsung Galaxy S23 FE
• Samsung Galaxy Z Fold 5
• Samsung Galaxy Z Flip 5
• Samsung Galaxy Tab S9, S9 Plus, and S9 Ultra
• Google Pixel 8 and Pixel 8 Pro
• Google Pixel 7 and Pixel 7 Pro
• Google Pixel 6 and Pixel 6 Pro
• Google Pixel 7a
• Google Pixel 6a
• Google Pixel Fold
• Google Pixel Tablet

What Types of Math Problems Can Google Circle Solve?

Currently, Google Circle is particularly helpful with math word problems. These can be some of the trickiest problems to tackle, as they require translating real-world situations into mathematical terms.

Google Circle excels at helping students understand the key information in a word problem, identify the relevant mathematical concepts, and apply the appropriate formulas to reach the solution.

Here are some specific areas where Google Circle can be helpful:

• Algebra: Solve for variables, manipulate equations, and understand linear relationships.
• Geometry: Calculate areas, and volumes, and understand geometric shapes and their properties.
• Statistics: Analyze data sets, calculate measures of central tendency, and interpret graphs.

How to Solve Math Probelm Using Google Search

Step 1: activate circle to search.

On your Android smartphone or tablet, activate the Circle to Search feature.

Step 2: Circle the Problem

Find the math problem you’re stuck on and circle it on your screen. This prompts the feature to focus on that particular problem.

Step 3: Get Step-by-Step Instructions

The feature breaks down the problem and provides step-by-step instructions to solve it

Google Circle Vs Photomath Vs Socratic by Google

Tips for getting the most out of google circle.

• Practice makes perfect: Don’t just rely on Google Circle for the answer. Use it as a guide to understand the steps and then try solving similar problems on your own. This will solidify your learning and build your confidence.
• Focus on understanding, not just answers: While getting the right answer is important, the true power of Google Circle lies in its explanations. Pay attention to why each step is taken and how the concepts are applied.
• Don’t be afraid to experiment: Google Circle can handle various math problems. Try circling different parts of a word problem to see how it breaks down the information and guides you toward the solution.
• Take notes: As Google Circle explains the steps, jot down important points or formulas for future reference. This will create a personal study guide tailored to your learning needs.
• Combine it with other resources: Google Circle is a powerful tool, but it shouldn’t be your only resource. Use it alongside your textbook, class notes, or online tutorials for a well-rounded understanding.

Google Circle is a new tool designed to help students to become independent math problem solvers. By using its step-by-step guidance and clear explanations, you can transform your approach to math, gaining a deeper understanding of concepts and building the confidence to tackle even the most challenging problems. So, next time you’re stuck on a math question, don’t hesitate to circle it with Google Circle and watch your problem-solving skills soar!

Google Circle for Students – FAQs

Is google circle available on iphones.

Unfortunately, as of now, Google Circle is only available on Android devices within the Google Search app. However, there might be plans for future expansion to other platforms.

Is Google Circle safe to use?

Google Circle is a feature within the official Google Search app, making it a safe and reliable tool. However, it’s always a good practice to be mindful of the information you share online and only use it for educational purposes.

What if Google Circle can’t solve my math problem?

While Google Circle is constantly learning and expanding its capabilities, it might encounter problems it can’t handle yet. In such cases, don’t hesitate to consult your teacher, a tutor, or online resources for alternative solutions.

Will Google Circle do my math homework for me?

Google Circle is not a magic shortcut. It’s designed to be a learning tool. While it can guide you through the problem-solving process, it’s still important for you to understand the steps and concepts involved.

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AIMS Mathematics

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Modified mildly inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and nonexpansive fixed point problems

• Francis Akutsah 1 , 
• Akindele Adebayo Mebawondu 1,2 ,  ,  , 
• Austine Efut Ofem 1 , 
• Reny George 3 ,  ,  , 
• Hossam A. Nabwey 3,4 , 
• Ojen Kumar Narain 1
• 1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
• 2. Mountain Top University, Prayer City, Ogun State Nigeria
• 3. Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
• 4. Department of Basic Engineering, Faculty of Engineering, Menoufia University, Shibin el Kom 32511, Egypt
• Received: 06 March 2024 Revised: 28 April 2024 Accepted: 09 May 2024 Published: 20 May 2024

MSC : 26A33, 34B10, 34B15

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This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm's weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments.

• subgradient extragradient ,
• mildly inertial ,
• equilibrium problem ,
• fixed point problem

Citation: Francis Akutsah, Akindele Adebayo Mebawondu, Austine Efut Ofem, Reny George, Hossam A. Nabwey, Ojen Kumar Narain. Modified mildly inertial subgradient extragradient method for solving pseudomonotone equilibrium problems and nonexpansive fixed point problems[J]. AIMS Mathematics, 2024, 9(7): 17276-17290. doi: 10.3934/math.2024839

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• This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ -->

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• © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/4.0 )

通讯作者: 陈斌, [email protected]

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• Ojen Kumar Narain

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What Is Tucker Carlson Even Doing?

His post–fox news years did not look promising. they’ve been bleak beyond imagination..

This is part of  Sly as Fox , a short series about the perils of underestimating Fox News in 2024.

It is always sort of heartwarming when a person fulfills their peak potential. Many of us never do so, after all, and instead spend our lives caught in the frustrating gap between ability and opportunity. When someone is able to bridge that gap and become their optimal self, it should count as cause for celebration. And so I am genuinely glad to see that former Fox News host Tucker Carlson is now doing the work that he was always meant to do: posting crackpot videos to the internet for an audience of angry loons.

It presumably came as the shock of Carlson’s life when, in April 2023, he was abruptly ousted from Fox News upon the sudden cancellation of his prime-time program there, Tucker Carlson Tonight . After all, Carlson was the unquestioned alpha of Fox’s talent roster, and his show often rated as the most popular prime-time show in all of cable television. He used his platform for many interesting purposes: to applaud political strongmen, to whitewash the events of Jan. 6, and to propagate a series of dark ethnonationalist theories that echoed themes long sounded in the least reputable corners of the internet. His formula was “Trumpism without Trump,” as the New York Times put it in a long 2022 profile, and it worked very well—right up until the day when his bosses, for whatever reason, decided that they had had enough.

While the justifications for his departure remain murky, the ramifications were as clear as day. Carlson was off the air, and because he was still under contract to Fox News—the contract reportedly runs through 2024 —he could not seek employment at another media outlet. Sidelined on the brink of a general-election campaign, facing sudden irrelevance upon the loss of his prominent platform, Carlson decided to follow in the footsteps of many deposed mainstream-media icons before him: He took his talents online.

Within weeks of leaving Fox News, Carlson announced that he would independently produce a version of his show for broadcast on Twitter, now “X,” a website that is not known for having shows. In the first episode , he pretty much picked up where he had left off on cable: carrying water for Vladimir Putin, sneering at mainstream-media groupthink, and calling Ukrainian president Volodymyr Zelensky, who is Jewish, “sweaty and rat-like, a comedian turned oligarch, a persecutor of Christians, a friend of BlackRock.” There’s, uh, a lot to parse there.

According to Twitter, Carlson’s debut logged 114 million views within a week of its release—a number that seems much less impressive when you realize that a “view” only means that a logged-in user watched at least 2 seconds of a video with at least half of the video player visible on the user’s screen. This inaugural episode was met with a cease-and-desist letter from Fox News, and while Carlson neither ceased nor desisted, it ultimately didn’t matter, because “Tucker on Twitter” did not sustain its initial momentum.

Carlson seemed to be learning the same hard lesson that many other would-be multiplatform demagogues have learned: The internet is already oversaturated with perfervid dog-whistling about shadowy globalist cabals. Carlson’s VDARE-lite punditry was a novelty on mainstream cable television. On Twitter, though, he was just another wingnut with a webcam. But rather than reading the room and deciding to sit back and count his money while waiting for his Fox contract to expire, Carlson instead chose to double down on his online ventures.

In December 2023, with seed capital from an “anti-woke” venture capital firm and in collaboration with his college roommate and co-founder of the Daily Caller, Neil Patel, and the former executive producer of his Fox News show, Justin Wells, Carlson launched a website called the “Tucker Carlson Network,” or TCN—not to be confused with the cable network TC M , which also likes to present things in black and white. (On Wednesday, Semafor reported that Wells is leaving TCN.)

I spent much of this past week poking around on TCN, which is perhaps best understood as a streaming service meant for the sorts of people who thought Heather Heyer had it coming. While I do not fully understand how this venture is allowable under the terms of Carlson’s Fox News contract, I suppose it’s possible that Fox took a look at TCN’s output and decided that there are some eyeballs that just aren’t worth competing for.

The website features various Tucker-centric programs covering various Tucker-approved topics, such as systemic racism against white Americans , the Biden administration’s alleged war on Christianity , and the many ways in which the QAnon Shaman got jobbed by the feds. Some TCN content is paywalled, and to access it you have to subscribe , for either $9 per month or $72 per year. Your subscription also buys you other perks, such as the opportunity to ask the host questions that he might subsequently choose to answer on camera, in an advice segment called Ask Tucker . (“I want to discipline my kids but I’m worried about turning into Tony Fauci . What do I do?”) The network’s logo resembles a “red pill,” as in the metaphorical medicine that alt-right podcasters urge their listeners to swallow so that they might stop feeling bad about being bigoted incels.

TCN’s shows are all pretty similar. Carlson opines on current events and conducts interviews with an ideologically predictable roster of guests, such as mixed martial artist and alleged cultural cancelee Gina Carano ; former Times reporter turned vaccine skeptic Alex Berenson ; Joe Biden sexual-assault accuser Tara Reade ; and Twitter user Catturd . He also interviewed the notorious mid-2000s internet jackass Tucker Max , whom Carlson inaccurately described as being a “beautiful prose stylist.” (I suppose I should not be surprised that America’s two most irritating Tuckers know each other.) Carlson also interviews heavier hitters, such as Andrew and Tristan Tate , the sibling social-media influencers who are each currently facing rape and human trafficking charges in Romania; Jeffrey Epstein’s brother Mark , who refused to appear on camera; and Vladimir Putin, whom Carlson traveled to Russia to meet earlier this year, and who subsequently disparaged the host for conducting a softball interview.

Putin was a big “get” for Carlson, no doubt about it, even if the interview made the host look more like a credulous tourist than a serious journalist. And yet most of Carlson’s guests on TCN are neither as prominent nor as important as Putin. The host isn’t sitting down with world leaders every week; he’s sitting down with the sorts of guests whom you can also find on plenty of other alt-right online shows and podcasts. After poking around the Tucker Carlson Network for the better part of a week, I couldn’t help but wonder: Who, exactly, is the website for?

Earlier in this column I speculated that Carlson’s audience was made up of angry loons—but as I already mentioned, there’s already a lot of competition online for these people’s attention, and many of these outlets do not require said loons to spend $9 per month to get their cortisol fix. “The whole vision is to create the next media company that is purpose-built for the 2020s and 2030s,” one of Carlson’s financiers told the Wall Street Journal . The pitch seemed somehow both ominous and empty, much like many of Carlson’s monologues. I might not be an anti-woke VC, but I do know enough about media to say that actual next-generation media companies generally aren’t built around middle-aged race-baiters interviewing cranks and weirdos from their paneled living room. That’s a podcast, not a viable startup.

Podcasts and online shows can be lucrative, that’s for sure, and perhaps Carlson’s intention is simply to bigfoot some of the other alt-right commentators who have staked their claims in that space. But to play in the same sandbox as people such as Joe Rogan and Alex Jones—in a recent appearance on Rogan’s show, Carlson implied that Jones may be some sort of prophet—Carlson’s going to have to keep matching or exceeding them thematically. The farther he goes over into true wingnut territory, the harder it becomes to build an actual media company around him.

What’s Carlson’s endgame here? A March interview that Carlson conducted with ex-CNN guy and current NewsNation host Chris Cuomo perhaps offers a clue. It ran for almost two hours, and while I confess to only skipping around in it instead of watching the whole thing, the unmistakable impression I got from the conversation is that these are two guys who really miss their old platforms. Seen in that context, you begin to understand that the Tucker Carlson Network is primarily meant for Tucker Carlson himself. It’s a vanity project designed to keep the host busy as he counts down the days to when he can return to cable once again.

For now, though, he’s working hard to remake his cable persona for the needs and desires of the alt-right internet. Although a new program called The Tucker Carlson Show just launched this week—Carlson’s first two guests were podcaster Dave Smith and ayahuasca enthusiast Aaron Rodgers , who announced , inaccurately, that Carlson’s interview with Putin had been “awesome”—TCN’s flagship show up to now has been Tucker Carlson Uncensored , in which the host monologues a bit and then generally chats with a guest. While the show’s title implies that the host was in fact censored in his previous job, the Tucker Carlson Uncensored episodes I watched were all thematically similar to his Fox News output.

If there is a discernable difference between this new, free-to-speak-his-mind version of Carlson and his ostensibly muzzled predecessor, I suppose it is that Uncensored Tucker hardly even bothers to camouflage his true opinions and intentions. The Ukraine skepticism and careful Putin apologism that pervaded the last year or so of his Fox show has become a full-fledged effort to portray Ukraine as the real enemy . And while Carlson spent countless hours on Fox News fanning the flames of xenophobic nativism, on Uncensored he’s doing stuff like interviewing white nationalist Lydia Brimelow. You get the picture. Carlson is no longer wasting any time trying to hide the ball, which I suppose is why I’m happy for him. In becoming an internet video guy, Carlson has emerged fully from the shadows and stepped fully into the unflattering light of his own biases. There is something to be said for self-actualization—if only so that the rest of us can see who someone’s true self was all along.