problem solving angles in polygons

Angles of Polygons

In these lessons, we will learn

• how to calculate the sum of interior angles of a polygon using the sum of angles in a triangle
• the formula for the sum of interior angles in a polygon
• how to solve problems using the sum of interior angles
• the formula for the sum of exterior angles in a polygon
• how to solve problems using the sum of exterior angles.

All the polygons in these lessons are assumed to be convex polygons .

Related Pages Polygons Quadrilaterals Cyclic Quadrilaterals More Geometry Lessons

The following diagrams give the formulas for the sum of the interior angles of a polygon and the sum of exterior angles of a polygon. Scroll down the page if you need more examples and explanation.

Sum Of Interior Angles Of A Polygon

We first start with a triangle (which is a polygon with the fewest number of sides). We know that

The sum of interior angles in a triangle is 180°.

This is also called the Triangle Sum Theorem. Click here if you need a proof of the Triangle Sum Theorem.

Next, we can figure out the sum of interior angles of any polygon by dividing the polygon into triangles. We can separate a polygon into triangles by drawing all the diagonals that can be drawn from one single vertex.

In the quadrilateral shown below, we can draw only one diagonal from vertex A to vertex B. So, a quadrilateral can be separated into two triangles.

The sum of angles in a triangle is 180°. Since a quadrilateral is made up of two triangles the sum of its angles would be 180° × 2 = 360°

The sum of interior angles in a quadrilateral is 360º

A pentagon (five-sided polygon) can be divided into three triangles. The sum of its angles will be 180° × 3 = 540°

The sum of interior angles in a pentagon is 540°.

A hexagon (six-sided polygon) can be divided into four triangles. The sum of its angles will be 180° × 4 = 720°

The sum of interior angles in a hexagon is 720°.

Formula For The Sum Of Interior Angles

We can see from the above examples that the number of triangles in a polygon is always two less than the number of sides of the polygon. We can then generalize the results for a n-sided polygon to get a formula to find the sum of the interior angles of any polygon.

The following diagram shows the formula for the sum of interior angles of an n-sided polygon and the size of an interior angle of a n-sided regular polygon. Scroll down the page for more examples and solutions on the interior angles of a polygon.

Example: Find the sum of the interior angles of a heptagon (7-sided)

Solution: Step 1: Write down the formula (n - 2) × 180°

Step 2: Plug in the values to get (7 - 2) × 180° = 5 × 180° = 900°

Answer: The sum of the interior angles of a heptagon (7-sided) is 900°.

Example: Find the interior angle of a regular octagon.

Answer: Each interior angle of an octagon (8-sided) is 135°.

Worksheet using the Formula for the Sum of Interior Angles

How to find the sum of the interior angles of any polygon using triangles and then derive the generalized formula?

Problems Using The Sum Of Interior Angles

How to find a missing angle using the sum of interior angles of a polygon?

How to use the sum of interior angles to write an equation and solve for the unknown? Write an equation and solve for the unknown. Substitute your answer into each expression to determine the measure of the angle. Give reasons for your answers.

Formula For The Sum Of Exterior Angles

The sum of exterior angles of any polygon is 360°.

The exterior angle of a regular n-sided polygon is 360°/n

Worksheet using the formula for the sum of exterior angles

Worksheet using the formula for the sum of interior and exterior angles

How to find the sum of the exterior angles and interior angles of a polygon? Every convex polygon has interior and exterior angles. The interior angles are inside the polygon formed by the sides. The exterior angles form a linear pair with the interior angles.

Example: Determine the measure of each exterior and interior angle of a regular polygon.

Problems using the sum of exterior angles

The following video shows a problem involving the sum of exterior angles of a polygon.

Example: A regular polygon has an exterior angle that measures 40°. How many sides does the polygon have?

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Angles In Polygons

Here we will learn about angles in polygons including how to calculate angles in polygons using a variety of methods and an overview of interior and exterior angles

There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are angles in polygons?

Angles in polygons relate to the interior and exterior angles of regular and irregular polygons.

Interior angles are the angles within a polygon made by two sides. We can calculate the sum of the interior angles of a polygon by subtracting 2 from the number of sides and then multiplying by 180º .

Step-by-step guide: Interior angles of a polygon

Exterior angles are the angles between a polygon and the extended line from the next side. The sum of the exterior angles of a polygon is always equal to 360º .

Step-by-step guide: Exterior angles of a polygon

A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines. ‘Poly’ comes from the greek for ‘many’ whilst ‘gon’ means ‘angles’. You will be familiar with many types of polygons such as triangle, rectangle and pentagon.

Regular polygons have all angles are that are equal in size and all sides that are equal in length.

Irregular polygons have angles that are not equal in size and sides that are not equal in length.

What are angles in polygons

Sum of the interior angles of a polygon

The ‘ sum of interior angles ’ of a polygon means finding the total of all the angles in a polygon. This is the key step in helping us solve many problems involving angles in polygons.

We know that the sum of all the angles in a triangle is equal to 180º , but what about angles in a quadrilateral? A regular pentagon? Or even an irregular octagon?

Sum of the angles in a triangle :

We know that the three angles in any triangle add up to 180º .

Mathematically we would say:

“The sum of interior angles for a triangle is 180 degrees” .

Step-by-step guide: Angles in a triangle

Sum of the angles in a quadrilateral :

A quadrilateral is a four sided shape. We can ‘split’ a quadrilateral into two triangles by drawing a line from one corner to an opposite one.

If the sum of interior angles one triangle is 180º , then the sum of the interior angles of two triangles is 180º × 2 = 360º .

So the sum of the interior angles of quadrilateral is 360º .

Step-by-step guide: Angles in a quadrilateral

Using our knowledge of triangles we can find the sum of the interior angles of any polygon by splitting it into triangles.

Step-by-step guide: Angles in a pentagon

Step-by-step guide: Angles in a hexagon

Key relationships

Interior and exterior angles add up to 180 0 .

Interior and exterior angles lie on a straight line. This means that when added together they will equal 180º .

This is useful when working on more complex questions.

The number of triangles created inside a shape is always 2 lower than the number of sides.

Here is a list of all the polygons we will work with in this lesson:

Notice how the number of triangles created is always 2 lower than the number of sides .

The sum of the interior angles of a polygon depends on how many sides it has, not what it looks like.

Despite these two decagons ( 10 sided shapes) looking very different, the sum of their interior angles is the same.

How to find the sum of the interior angles of a polygon

In order to find the sum of interior angles for any polygon you should:

Identify how many sides the polygon has.

Identify if the polygon is regular or irregular.

• If possible work out how many triangles could be created within the polygon by drawing lines from one vertex to all the other vertices.
• Multiply the number of triangles by 180 to calculate the sum of the interior angles.

State your findings e.g. sides, regular/irregular, the sum of interior angles.

How to find the sum of the interior angles for a polygon

Angles in polygons worksheet

Get your free angles in polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Angles in polygons examples

Example 1: irregular quadrilateral.

Find the sum of interior angles for this polygon.

This polygon has four sides.

2   Identify if the polygon is regular or irregular.

This polygon is irregular as all the sides are not of equal length.

3 If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

The polygon can be broken up into two triangles.

4 Multiply the number of triangles by 180 to get the sum of the interior angles.

5 State your findings e.g. sides, regular/irregular, the sum of interior angles.

The polygon is a irregular quadrilateral (specifically called a parallelogram as both opposite sides are parallel) with a sum of interior angles of 360º .

Example 2: regular decagon

This polygon has 10 sides.

This polygon is regular as all the sides are of equal length and the angles are of equal size.

If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

The polygon can be broken up into eight triangles.

Multiply the number of triangles by 180 o to get the sum of the interior angles.

The polygon is a regular 10 sided polygon (decagon) with a sum of interior angles of 1440º .

Example 3: irregular pentagon with concave angle

This polygon has five sides.

This polygon is irregular as all the sides are not of equal length and the angles are not of equal size

The polygon can be broken up into three triangles.

The polygon is a irregular five sided polygon (pentagon) with a sum of interior angles of 540º .

Example 4: complex polygon

This polygon has 10 sides

This polygon is irregular as all the angles are not of equal size.

Notice how we had to do this differently to previous examples. But we have not created any new angles because our lines do not cross.

The polygon is a irregular 10 sided polygon with a sum of interior angles of 1440º .

STOP AND THINK: Notice how this is the same sum of interior angles for example 2. This is because the polygon has the same number of sides .

Example 5: finding an exterior angle

  What is the size of one of the exterior angles of an equilateral triangle?

The sum of interior angle in a triangle is 180º .

An equilateral triangle has 3 angles of equal size.

Therefore each interior angle is 60º because 180 ÷ 3 = 60 .

Therefore one exterior angle is equal to 180 – 60 because the interior angle and exterior angle of a polygon lie on a straight line.

Therefore one exterior angle is 120º .

Example 6: finding the sum of interior angles of larger polygons

What is the sum of the interior angles for a 30 sided polygon?

To work out the number of triangles a polygon can be split into we subtract 2 .

A 30 sided polygon can be split into 30 − 2 = 28 triangles.

Therefore the sum of its interior angles can be found by 28 × 180º = 5040º .

The sum of interior angles of a 30 sided polygon is 5040º .

This will be looked at in more detail in the lesson on interior angles in a polygon .

Example 7: finding the sum of interior angles of a complex polygon

This polygon has 6 sides. The sides are not all the same length so this polygon is an irregular hexagon.

To work out the number of triangles a polygon can be split into we subtract 2 from the number of sides it has.

A 6 sided polygon can be split into 6 − 2 = 4 triangles.

Therefore the sum of its interior angles can be found by 4 x 180º = 720º .

The sum of interior angles in a hexagon is 720º .

Common misconceptions

• Miscounting the number of sides
• Misidentifying if a polygon is regular or irregular
• Incorrectly assuming all the angles are the same size
• Crossing lines when drawing the triangles, this creates false interior angles

Practice angles in polygons questions

1. Is a square a regular or irregular polygon?

All the side lengths are the same, and all angles are right angles, hence a square is a regular polygon.

2. Is a semi circle a polygon?

A semicircle has a side which is not straight, so it is not a polygon.

3. What is the sum of interior angles for a triangle?

180^{\circ} 

360^{\circ} 

270^{\circ} 

The angles in a triangle add up to 180^{\circ}.

4. What is the sum of interior angles for a regular hexagon?

720^{\circ}

540^{\circ}

360^{\circ}

1080^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A regular hexagon can be divided into 4 triangles, and four lots of 180 is 720 .

5. What is the sum of interior angles for an irregular hexagon?

We know the angles in a triangle add up to 180^{\circ} . An irregular hexagon can be divided into 4 triangles, and four lots of 180 is 720 .

6. What is the sum of interior angles for a regular 12 sided polygon?

2700^{\circ}  

2160^{\circ}

1800^{\circ}  

1080^{\circ}  

We know the angles in a triangle add up to 180^{\circ} . A 12 sided polygon can be divided into 10 triangles, and ten lots of 180 is 1800 .

7. What is the sum of interior angles for a regular 25 sided polygon?

4500^{\circ}

4140^{\circ}

9000^{\circ}

3600^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A 25 sided polygon can be divided into 23 triangles, and 23 lots of 180 is 4140 .

Angles in polygons GCSE questions

1.  Each exterior angle of a regular polygon is 15^{\circ} . Work out the number of sides the polygon has.

360 \div 15

            (1)

2.  Each of the interior angles of a regular polygon is 140^{\circ} . Show that this polygon has 9 sides

Exterior angle = 40 seen or implied

3.  In a regular polygon each exterior angle is 18^{\circ} . Find the sum of interior angles for this polygon

Learning checklist

You have now learned how to:

• Use conventional terms for geometry e.g. regular or irregular
• Derive and apply the properties and definitions of: special types of quadrilaterals
• Knowing names and properties of polygons
• Calculate the sum of interior angles for a regular polygon
• Calculate the sum of interior angles for a irregular polygon

The next lessons are

• Angle rules
• Angles in parallel lines
• How to calculate volume

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Angles in Polygons (Challenges – Part 1)

Subject: Mathematics

Age range: 11-14

Resource type: Lesson (complete)

Last updated

17 January 2019

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Key learning points.

• In this lesson, we will learn about how to problem solve with angles in polygons, along with learning notations for referring to angles.

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Geometry (all content)

Course: geometry (all content)   >   unit 3.

• Sum of interior angles of a polygon
• Sum of the exterior angles of a polygon

Angles of a polygon

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

• Mathematicians
• Math Lessons
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• Angles in Polygons – Explanation & Examples

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Interior angle of polygons

Exterior angle of polygons, practice questions, angles in polygons – explanation & examples.

How to differentiate them then? ANGLES!

The simplest example is that both rectangle and a parallelogram have 4 sides each, with opposite sides are parallel and equal in length. The difference lies in angles, where a rectangle has 90-degree angles on its all 4 sides while a parallelogram has opposite angles of equal measure.

In this article, you will learn:

• How to find the angle of a polygon?
• Interior angles of a polygon.
• Exterior angles of a polygon.
• How to calculate the size of each interior and exterior angle of a regular polygon.  

How to Find the Angles of a Polygon?

We know that a polygon is a two-dimensional multi-sided figure made up of straight-line segments . The sum of angles of a polygon is the total measure of all interior angles of a polygon.

Since all the angles inside the polygons are the same. Therefore, the formula for finding the angles of a regular polygon is given by;

Sum of interior angles = 180° * (n – 2)

Where n = the number of sides of a polygon.

• Angles of a Triangle:

a triangle has 3 sides, therefore,

Substitute n = 3 into the formula of finding the angles of a polygon.

= 180° * (3 – 2)

• Angles of a Quadrilateral:

A quadrilateral is a 4-sided polygon, therefore,

By substitution,

sum of angles = 180° * (n – 2)

= 180° * (4 – 2)

• Angles of a Pentagon

A pentagon is a 5 – sided polygon.

Substitute.

=180° * (5 – 2)

• Angles of an octagon.

An Octagon is an 8 – sided polygon

= 180° * (8 – 2)

Angles of a Hectagon:

a Hectagon is a 100-sided polygon.

= 180° * (100 – 2)

= 180° * 98

The interior angle is an angle formed inside a polygon, and it is between two sides of a polygon.

The number of sides in a polygon is equal to the number of angles formed in a particular polygon. The size of each interior angle of a polygon is given by;

Measure of each interior angle = 180° * (n – 2)/n

where n = number of sides.

• Size of the interior angle of a decagon.

A decagon is a 10 -sided polygon.

Substitution.

= 180° * (10 – 2)/10

= 180° * 8/10

• Interior angle of a Hexagon.

A hexagon has 6 sides. Therefore, n = 6

Measure of each interior angle =180° * (n – 2)/n

= 180° * (6 – 2)/6

= 180° * 4/6

• Interior angle of a rectangle

A rectangle is an example of a quadrilateral (4 sides)

=180° * (4 – 2)/4

=180° * 1/2

• Interior angle of a pentagon.

A pentagon is composed of 5 sides.

The measure of each interior angle =180° * (5 – 2)/5

=180° * 3/5

The exterior angle is the angle formed outside a polygon between one side and an extended side. The measure of each exterior angle of a regular polygon is given by;

The measure of each exterior angle =360°/n, where n = number of sides of a polygon.

• Exterior angle of a triangle:

For a triangle, n = 3

Measure of each exterior angle = 360°/n

• Exterior angle of a Pentagon:

NOTE: The interior angle and exterior angle formulas only work for regular polygons. Irregular polygons have different interior and exterior measures of angles.

Let’s look at more example problems about interior and exterior angles of polygons.

The interior angles of an irregular 6-sided polygon are; 80°, 130°, 102°, 36°, x°, and 146°.

Calculate the size of angle x in the polygon.

For a polygon with 6 sides, n = 6

the sum of interior angles =180° * (n – 2)

= 180° * (6 – 2)

Therefore, 80° + 130° + 102° +36°+ x° + 146° = 720°

494° + x = 720°

Subtract 494° from both sides.

494° – 494° + x = 720° – 494°

Find the exterior angle of a regular polygon with 11 sides.

The measure of each exterior angle= 360°/n

The exterior angles of a polygon are; 7x°, 5x°, x°, 4x° and x°. Determine the value of x.

Sum of exterior =360°

7x° + 5x° + x° + 4x° + x° =360°

Divide both sides by 18.

x = 360°/18

Therefore, the value of x is 20°.

What is the name of a polygon whose interior angles are each 140°?

Size of each interior angle = 180° * (n – 2)/n

Therefore, 140° = 180° * (n – 2)/n

Multiply both sides by n

140°n =180° (n – 2)

140°n = 180°n – 360°

Subtract both sides by 180°n.

140°n – 180°n = 180°n – 180°n – 360°

-40°n = -360°

Divide both sides by -40°

n = -360°/-40°

Therefore, the number of sides is 9 (nonagon).

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Angles in Polygons

October 16, 2019.

There are two key learning points when solving problems with angles in polygons.  The first is to understand why all the exterior angles of a polygon have a sum of 360°.  The second is to understand the interior and exterior angles appear on the same straight line.

Students can be told these two facts and attempt to internalise them by repeated practice.  However, this would be an opportunity missed.  Exploring the properties of polygons through construction can be a fantastic discovery of patterns and geometric properties.

Polygon or not.  Regular or not.

I begin the lesson by asking students to categorise a selection of 2D shapes into groups.  This can be done through discussion or in their books.  I don’t use mini-whiteboards this lesson as the student’s desks will be busy with rulers, pairs of compasses, plain paper and protractors.

Through the class discussion I draw out any misconceptions about how to define a regular polygon.  This is essential knowledge for the remainder of the lesson.

I explain to the students that the aim of today’s lesson is to discover some (I do not say how many) angle properties of polygons.  To make this as simple as possible all the polygons will need to be regular.  Later, we will extend these properties to calculate other facts about polygons.

Constructing regular polygons

We begin by constructing a regular triangle within a circle.  I ask all students to construct their own circles using a pair of compasses.  This may seem a simple task, however it often proves to be the most challenging part of the lesson as compasses often slip and students can lack dexterity. 

The class and I construct the regular triangle and quadrilateral together.  When the polygon has been constructed I ask students to extend the edges to make a windmill shape.  Later, this will help to visualise the exterior angles.  The windmill analogy proves helpful to visualise the sum of the exterior angles as 360°.

Click here to view the video.

Having constructed the triangle and square I leave the students to work independently to construct a pentagon and hexagon.  They use the constructions to measure the interior and exterior angles.

About 40 minutes into the lesson most students have completed up to a hexagon.  At this point I stop the class to remind them of the aim of the lesson. 

I explain to the class that to identify a pattern it is helpful to display our results within a table.  I use student’s results to partially complete the table as this provides an opportunity to discuss accuracy in the construction or measurement of angles.

I ask the students to discuss any patterns they notice and to use these patterns to predict the likely results for the next four polygons. The predictions can be tested, and if needed rethought by constructing the heptagon and octagon.

Discovering Angles in Polygons

In the plenary I ask students to discuss the patterns they have noticed and how they can be explained.  All the students identify the sum of the exterior angles as 360°.  When I challenged them to explain why the majority refer back to the initial circle we constructed.  I ask them to think deeper and with a greater focus on the windmills created by the extended edges.   Eventually, word begins to spread that the all exterior edges rotate around a circle which is why they have a sum of 360°.

I ask the class to look for another pattern.  A short while later a student shares the exterior and interior angles lie along a straight line which is why they add to 180°.

I challenge the class to check whether these two patterns work with their own results.  If they do not can we explain why not?  Students are quick to point out any instances where they do not work can be explained through inaccurate measurements or constructions.

Applying the properties of angles in polygons

Constructing the polygons, measuring the interior and exterior angles, and completing the table of results typically takes a full 60 minute lesson and homework.  In the next lessons students use the properties of regular polygons to solve a range of complex problems involving multiple angle properties.

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My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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Polygons: Formula and Examples

Exterior angles and interior angles, interior angle sum theorem, what is true about the sum of interior angles of a polygon .

The sum of the measures of the interior angles of a convex polygon with n sides is $ (n-2)\cdot180^{\circ} $

What is the total number degrees of all interior angles of a triangle ?

You can also use Interior Angle Theorem :$$ (\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ} $$

What is the total number of degrees of all interior angles of the polygon ?

360° since this polygon is really just two triangles and each triangle has 180°

You can also use Interior Angle Theorem :$$ (\red 4 -2) \cdot 180^{\circ} = (2) \cdot 180^{\circ}= 360 ^{\circ} $$

What is the sum measure of the interior angles of the polygon (a pentagon) ?

Use Interior Angle Theorem :$$ (\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$

What is sum of the measures of the interior angles of the polygon (a hexagon) ?

Video Tutorial on Interior Angles of a Polygon

Definition of a Regular Polygon:

Examples of regular polygons.

Measure of a Single Interior Angle

What about when you just want 1 interior angle?

In order to find the measure of a single interior angle of a regular polygon  (a polygon with sides of equal length and angles of equal measure) with n sides, we calculate the sum interior angles or $$ (\red n-2) \cdot 180 $$ and then divide that sum by the number of sides or $$ \red n$$.

$ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} $

So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons.

To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $

Finding 1 interior angle of a regular Polygon

What is the measure of 1 interior angle of a regular octagon?

Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle

Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle

Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?

Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle

Challenge Problem

What is the measure of 1 interior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!

Consider, for instance, the ir regular pentagon below.

You can tell, just by looking at the picture, that $$ \angle A    and    \angle B $$ are not congruent .

The moral of this story- While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measure--except when the polygon is regular .

How about the measure of an exterior angle?

Formula for sum of exterior angles: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.

Measure of a Single Exterior Angle

$$ \angle1 + \angle2 + \angle3 = 360° $$

$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$

$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$

Practice Problems

Calculate the measure of 1 exterior angle of a regular pentagon?

Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle

What is the measure of 1 exterior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360 , you can only use formula to find a single exterior angle if the polygon is regular!

Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A \text{ and } and \angle B $$ are not congruent..

Determine Number of Sides from Angles

It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.

If each exterior angle measures 10°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

If each exterior angle measures 20°, how many sides does this polygon have?

If each exterior angle measures 15°, how many sides does this polygon have?

If each exterior angle measures 80°, how many sides does this polygon have?

When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 sides.

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PROBLEMS INVOLVING INTERIOR ANGLES OF A POLYGON

Find the value of x in the following problems, give brief reasons for your answers.

Problem 1 :

Number of sides of the polygon given above = 5.

Sum of interior angles of given polygon = (n - 2)180°

= (5 - 2)180°

108° + 108° + 108° + 108° + x° = 540°

432 + x = 540

Subtract 432 from both sides.

x = 108

Problem 2 :

Number of sides of the polygon given above = 6

Sum of interior angles of given polygon = (n - 2)180°

= (6 - 2)180°

= (4)180°

= 720°

130° + x° + 118° + 112° + 120° + 90° = 720°

570 + x = 720

Subtract 570 from both sides.

x = 150

Problem 3 :

Number of sides = 5

= (3)180°

2x° + 2x° + 2x° + 2x° + x° = 540°

9x = 540

Divide both sides by 9.

x = 60

Problem 4 :

Number of sides = 6

120° + 150° + x° + x° + x° = 720°

270 + 3x = 720

Subtract 270 from both sides.

3x = 450

Divide both sides by 3.

x = 150

Problem 5 :

Number of sides = 7

= (7 - 2)180°

= (5)180°

= 900°

60° + x° + x° + x° + x° + 2x° + 90° = 720°

150 + 6x = 720

Subtract 150 from both sides.

6x = 570

Divide both sides by 6.

x = 95

Problem 6 :

A regular polygon has interior angle of 156 ° . How many sides does the polygon have?

Polygon is made up of a finite number of straight lines which are connected to form a closed polygonal circuit.

Sum of interior and exterior angle = 180°

Interior angle = 156°

Exterior angle = 180° - 156° = 24°

Number of sides of polygon = 360° /exterior angle

= 360° /24°

So, the polygon has 15 sides.

Problem 7 :

Five identical isosceles triangles are put together to form a regular pentagon.

a) Explain why 5x = 360.

b) Find x and y.

In each triangle, sum of interior angles = 180°

x° + y° + y° = 180°

x + 2y = 180 -----(1)

5x = 360

Divide both sides by 5.

x = 72

Substitute x = 72 in (1).

72 + 2y = 180

Subtract 72 from both sides.

2y = 108

Divide both sides by 2.

y = 54

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Resources tagged with: Triangles

There are 109 NRICH Mathematical resources connected to Triangles , you may find related items under Angles, polygons, and geometrical proof .

Name That Triangle!

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

Triangle or No Triangle?

Here is a selection of different shapes. Can you work out which ones are triangles, and why?

Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Paper Patchwork 2

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

Isosceles Seven

Is it possible to find the angles in this rather special isosceles triangle?

Constructing Triangles

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Walking Round a Triangle

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

Data Shapes

Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

Sorting Logic Blocks

This activity focuses on similarities and differences between shapes.

Can you each work out what shape you have part of on your card? What will the rest of it look like?

Repeating Patterns

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

What does the overlap of these two shapes look like? Try picturing it in your head and then use some cut-out shapes to test your prediction.

Inside Triangles

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

Matching Triangles

Can you sort these triangles into three different families and explain how you did it?

Fraction Fascination

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

More Transformations on a Pegboard

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Triangle Animals

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Triangle in a Triangle

Can you work out the fraction of the original triangle that is covered by the inner triangle?

Property Chart

A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

Shapely Pairs

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

Nine-pin Triangles

How many different triangles can you make on a circular pegboard that has nine pegs?

Triangles All Around

Can you find all the different triangles on these peg boards, and find their angles?

Isosceles Triangles

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Tangram Tangle

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Transformations on a Pegboard

How would you move the bands on the pegboard to alter these shapes?

Rectangle Tangle

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?

Egyptian Rope

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Of All the Areas

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Chain of Changes

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Colouring Triangles

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Let's Investigate Triangles

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

Kissing Triangles

Determine the total shaded area of the 'kissing triangles'.

Three Way Split

Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

Area I'n It

Triangle ABC has altitudes h1, h2 and h3. The radius of the inscribed circle is r, while the radii of the escribed circles are r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1 + 1/r2 + 1/r3 .

Part the Polygons

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

Three Fingers and a Loop of String

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.

Seven Sticks

Explore the triangles that can be made with seven sticks of the same length.

Cut and Make

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

Tessellating Triangles

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

Cutting Corners

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

How many triangles can you make on the 3 by 3 pegboard?

Interior and Exterior Angles of Regular Polygons Word Problems

The angles that lie inside a shape, generally a polygon, are said to be interior angles .

An exterior angle of a polygon is the angle that is formed between any side of the polygon and a line extended from the next side. Every polygon has interior and exterior angles. The exterior is the term opposite to the interior which means outside. Therefore exterior angles can be found outside the polygon. The sum of the exterior angles of any polygon is equal to 360°. Any flat shape or figure is said to have interior or exterior angles only if it is a closed shape.

To find the sum of all interior angles in a regular polygon: The sum of the interior angles of a polygon can be found by taking the number of sides (n) and subtracting 2. Then, multiply that number by 180.

Sum of interior angles = (n – 2) ∙ 180°

To find the measure of each interior angle in a regular polygon: 1. To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below. Sum of interior angles = (n – 2) ∙ 180°

2. Next, divide the sum of interior angles by the total number of angles the regular polygon has.

To find the measure of each exterior angle in a regular polygon: The measure of one of the exterior angles of a regular polygon can be found by dividing 360 degrees by the number of angles (n).

Bees build honeycombs with hexagonal cells. What is the measure of each interior angle of the cell?

To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below.

A hexagon has 6 sides, so: (6 – 2) ∙ 180 = 4 ∙ 180 = 720°

Since this is a regular hexagon, all of the angles are equal, so divide the sum of the interior angles by 6.

Practice Interior and Exterior Angles of Regular Polygons Word Problems

Practice Problem 1

Practice Problem 2

Practice Problem 3

Polygon – A closed figure formed by three or more segments called sides.​

Interior angle – An angle of a polygon formed by two of its side and is inside the polygon. ​​

Exterior angle – An angle formed by one side and the extension of the adjacent side. It is outside the polygon.

Pre-requisite Skills Classify Angles Drawing Angles Estimating Angles Angle Relationships Classify Triangles Angles Finding Angle Measures Complementary and Supplementary Angles Angles in Triangles

Related Skill Geometric Proof

Rivian just got a $5 billion investment from Volkswagen, and it could help Rivian solve one of its biggest problems

• Volkswagen will invest up to $5 billion in Rivian, an EV maker that's struggled with profitability.
• Rivian, which lost $1.4 billion in the first quarter, aims to develop a more affordable SUV. 
• Rivian's new $45,000 SUV, the R2, set to launch in 2026, targets mainstream customers.

Volkswagen just inked a deal to invest up to $5 billion in Rivian, the electric-vehicle maker that has, like its peers, struggled to turn a profit.

The deal, announced Tuesday, creates a joint venture focused on developing software to be used in both companies' cars. It also affords Rivian, which lost $1.4 billion in the first quarter, a financial cushion to continue developing a much cheaper SUV.

The partnership is expected to "lower cost per vehicle by increasing scale and speeding up innovation globally," the joint announcement said.

Software is one of Rivian's strengths, Goldman Sachs analysts noted in January — "a key part of the value proposition and monetization opportunity for Rivian."

But the company has struggled to sell more cars profitably. It's still just a small slice of the overall EV market. Rivian delivered nearly 14,000 cars in the first quarter, compared with Tesla's almost 390,000.

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The race to create cheaper EVs is heating up across the industry, as car makers confront an affordability problem that has narrowed the potential buyer pool.

Rivian's current SUV costs nearly $80,000. In March, the company introduced two smaller SUVs designed to reach more mainstream customers.

"These represent our future," CEO RJ Scaringe said of the new models.

One of the models, called the R2, is slated to launch in 2026 and will cost $45,000, about the same as Tesla's Model Y.

The R2 "will be foundational to Rivian's long-term growth and profit potential," the company said in its first-quarter earnings report.

Tesla has been teasing a $25,000 electric vehicle for years. Last year, Tesla's top engineers told investors that the company's next generation of electric models would cost 50% less to make . But the production road has been bumpy — last month, Tesla pulled back on plans to roll out a new manufacturing method for its electric cars, Reuters reported.

In April, Ford CEO Jim Farley said the company is working on new affordable EVs that could be priced as low as $25,000 to $30,000.

The Volkswagen news sent Rivian shares up 50% in after-hours trading.

Watch: What it's like to own a $20 million exotic car collection

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