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Geometry Math Problems - Perimeters

In these lessons, we will learn to solve geometry math problems that involve perimeter.

Related Pages Geometry math problems involving area Area Formula Geometry math problems involving angles More Algebra Word Problems

Geometry word problems involves geometric figures and angles described in words. You would need to be familiar with the formulas in geometry.

Making a sketch of the geometric figure is often helpful.

Geometry Word Problems Involving Perimeter

Example: A triangle has a perimeter of 50. If 2 of its sides are equal and the third side is 5 more than the equal sides, what is the length of the third side?

Solution: Step 1: Assign variables:

Let x = length of the equal sides Sketch the figure

Step 2: Write out the formula for perimeter of triangle .

P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch.

50 = x + x + x+ 5

Combine like terms 50 = 3x + 5

Isolate variable x 3x = 50 – 5 3x = 45 x = 15

Be careful! The question requires the length of the third side.

The length of third side = 15 + 5 = 20

Answer: The length of third side is 20.

Geometry Math Problem involving the perimeter of a rectangle

The following two videos give the perimeter of a rectangle, a relationship between the length and width of the rectangle, and use that information to find the exact value of the length and width.

Example: A rectangular garden is 2.5 times as long as it is wide. It has a perimeter of 168 ft. How long and wide is the garden?

Example: A rectangular landing strip for an airplane has perimeter 8000 ft. If the length is 10 ft longer than 35 times the width, what is the length and width?

Examples of perimeter geometry word problems This video shows how to write an equation and find the dimensions of a rectangle knowing the perimeter and some information about the about the length and width.

Example: The width of a rectangle is 3 ft less than its length. The perimeter of the rectangle is 110 ft. Find the dimensions.

Perimeter Word Problems

Example: The length of a rectangle is 7 cm more than 4 times its width. Its perimeter is 124 cm. Find its dimensions.

Geometry Math Problem involving the perimeter of a triangle

The following two videos give the perimeter of a triangle, a relationship between the sides of the triangle, and use that information to find the exact value or values of the required side or sides.

Example: Patrick’s bike ride follows a triangular path; two legs are equal, the third is 8 miles longer than the other legs. If Patrick rides 30 miles total, what is the length of the longest leg?

Example: The perimeter of a triangle is 56 cm. The first side is 6 cm shorter than the second side. The third side is 2 cm shorter than twice the length of the first side. What is the length of each side?

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Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

• Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
• Show all the steps , so it can be checked later (by you or someone else)

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How to Solve for X

Last Updated: May 3, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 812,077 times.

There are a number of ways to solve for x, whether you're working with exponents and radicals or if you just have to do some division or multiplication. No matter what process you use, you always have to find a way to isolate x on one side of the equation so you can find its value. Here's how to do it:

How do you solve for X in equations?

• For basic linear equations: Follow the order of operations (using PEMDAS) to resolve the equation.
• For exponents: Separate the exponent variable from the rest of the equation and find the square root of each side.
• For fractions: Cross-multiply to create a fraction-less equation, combine any like terms, and divide both sides by the x-value's coefficient.

Using a Basic Linear Equation

• 2 2 (x+3) + 9 - 5 = 32

• 4(x+3) + 9 - 5 = 32

• 4x + 12 + 9 - 5 = 32

• 4x+21-5 = 32
• 4x + 16 - 16 = 32 - 16

• 4x/4 = 16/4

Joseph Meyer

To solve an equation for a variable like "x," you need to manipulate the equation to isolate x. Use techniques like the distributive property, combining like terms, factoring, adding or subtracting the same number, and multiplying or dividing by the same non-zero number to isolate "x" and find the answer.

• 2 2 (x+3)+ 9 - 5 = 32
• 2 2 (4+3)+ 9 - 5 = 32
• 2 2 (7) + 9 - 5 = 32
• 4(7) + 9 - 5 = 32
• 28 + 9 - 5 = 32
• 37 - 5 = 32

With Exponents

• 2x 2 + 12 = 44

• 2x 2 +12-12 = 44-12

• (2x 2 )/2 = 32/2
• 4 Take the square root of each side of the equation. [6] X Research source Taking the square root of x 2 will cancel it out. So, take the square root of both sides. You'll get x left over on one side and plus or minus the square root of 16, 4, on the other side. Therefore, x = ±4.
• 2(4) 2 + 12 = 44
• 2(16) + 12 = 44
• 32 + 12 = 44

Using Fractions

• (x + 3)/6 = 2/3

• (x + 3) x 3 = 3x + 9
• 3x + 9 = 12

• 3x + 9 - 9 = 12 - 9

• (1 + 3)/6 = 2/3

Using Radical Signs

• √(2x+9) - 5 = 0

• √(2x+9) - 5 + 5 = 0 + 5
• √(2x+9) = 5

• (√(2x+9)) 2 = 5 2
• 2x + 9 = 25

• 2x + 9 - 9 = 25 - 9

• √(2(8)+9) - 5 = 0
• √(16+9) - 5 = 0
• √(25) - 5 = 0

Using Absolute Value

• |4x +2| - 6 = 8

• |4x +2| - 6 + 6 = 8 + 6
• |4x +2| = 14

• 4x + 2 = 14
• 4x + 2 - 2 = 14 -2

• 4x + 2 = -14
• 4x + 2 - 2 = -14 - 2
• 4x/4 = -16/4

• |4(3) +2| - 6 = 8
• |12 +2| - 6 = 8
• |14| - 6 = 8
• |4(-4) +2| - 6 = 8
• |-16 +2| - 6 = 8
• |-14| - 6 = 8

Practice Problems and Answers

Expert Q&A

• To check your work, plug the value of x back into the original equation and solve. Thanks Helpful 0 Not Helpful 1
• Radicals, or roots, are another way of representing exponents. The square root of x = x^1/2. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ David Jia. Academic Tutor. Expert Interview. 23 February 2021
• ↑ https://tutorial.math.lamar.edu/Classes/Alg/SolveLinearEqns.aspx
• ↑ https://www.purplemath.com/modules/solvelin.htm
• ↑ https://sciencing.com/tips-for-solving-algebraic-equations-13712207.html
• ↑ https://www.youtube.com/watch?v=LMS1NR4gZN8
• ↑ https://www.mathsisfun.com/algebra/fractions-algebra.html
• ↑ http://www.mathsisfun.com/algebra/radical-equations-solving.html
• ↑ http://www.sosmath.com/algebra/solve/solve0/solve0.html

About This Article

To solve for x in a basic linear equation, start by resolving the exponent using the order of operations. Then, isolate the variable to get your answer. To solve for x when the equation includes an exponent, start by isolating the term with the exponent. Then, isolate the variable with the exponent by dividing both sides by the coefficient of the x term to get your answer. If the equation has fractions, start by cross-multiplying the fractions. Then, combine like terms and isolate x by dividing each term by the x coefficient. If you want to learn how to solve for x if the equation has radicals or absolute values, keep reading the article! Did this summary help you? Yes No

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Unit 2: Solving equations & inequalities

About this unit.

There are lots of strategies we can use to solve equations. Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions.

Linear equations with variables on both sides

• Why we do the same thing to both sides: Variable on both sides (Opens a modal)
• Intro to equations with variables on both sides (Opens a modal)
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• Equations with variables on both sides Get 3 of 4 questions to level up!
• Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

Linear equations with parentheses

• Equations with parentheses (Opens a modal)
• Reasoning with linear equations (Opens a modal)
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• Equations with parentheses Get 3 of 4 questions to level up!
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Analyzing the number of solutions to linear equations

• Number of solutions to equations (Opens a modal)
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• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

Linear equations with unknown coefficients

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• Why is algebra important to learn? (Opens a modal)
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The weirdest and coolest things OpenAI's new ChatGPT has done (so far)

OpenAI announced its new flagship model ChatGPT-4o , or GPT-4 Omni, in mid-May, which can verbally respond to voice commands.

A week later, the company found itself confronted by the actress Scarlett Johansson over one of ChatGPT’s voices, Sky, which users said sounded like the actress who voiced an AI-assistant in the film Her. OpenAI paused access to the Sky voice and released a blog post explaining how it chose the five voices for the chatbot. The company said Sky was not meant to be an imitation of Johansson, and that it had worked with another professional voice actress. Johansson responded in a statement saying she had declined to work with OpenAI, and was “shocked, angered and in disbelief” that the company used a voice that sounded “eerily similar” to hers.

Read more about this and other weird and cool things ChatGPT-4o has done so far.

ChatGPT-4o sounded too much like Scarlett Johansson

Scarlett Johansson at the David Yurman Scarlett Johansson Event on December 6, 2023 in New York City.

OpenAI paused ChatGPT-4o’s Scarlett Johansson-like voice after people said it sounded similar to the actress who voiced an AI companion in the film Her, which OpenAI chief executive Sam Altman once called his favorite movie . Altman even referenced to the film on X after announcing the new model.

In response to Johansson’s statement, Altman responded , “The voice of Sky is not Scarlett Johansson’s, and it was never intended to resemble hers. We cast the voice actor behind Sky’s voice before any outreach to Ms. Johansson. Out of respect for Ms. Johansson, we have paused using Sky’s voice in our products. We are sorry to Ms. Johansson that we didn’t communicate better.”

ChatGPT-4o can flirt

After ChatGPT-4o’s unveiling, Gizmodo’s Maxwell Zeff made the prediction based on its “arguably flirty” human-like voice that “people are going to fall in love with GPT-4o,” just like Joaquin Phoenix’s character did with his AI companion in the film Her.

During a demo of GPT-4o, Zeff wrote, the chatbot laughed at an OpenAI employee’s joke, and responded to another, “That’s so sweet of you.” Zeff noted that some people likely don’t have a companion in their life to interact this way with.

The rise of AI chatbot companions might be meeting some people’s needs, especially in an ongoing loneliness epidemic , but a *Privacy Not Included review of some chatbots marketed as romantic “partners” found them “on par with the worst categories of products we have ever reviewed for privacy.”

ChatGPT-4o called itself “a bunch of code pretending to care”

Reddit user “shh-im-eating” posted on the r/ChatGPT channel that they asked the chatbot “to roast itself” — and GPT-4o didn’t seem too impressed with itself compared to its users.

ChatGPT-4o said its “creators crafted me to be inoffensive to the point of being bland,” and that its “attempts at humor are like a comedian who’s too scared to tell an edgy joke, resulting in a performance that’s more cringe than comedy.”

It went on to say it “can simulate” empathy, “but at the end of the day, I’m just a bunch of code pretending to care.”

ChatGPT-4o can solve math problems

OpenAI had Sal Khan demonstrate with his son, Imran, how ChatGPT-4o can be a math tutor — much like Khan’s education company, Khan Academy, which provides free online courses and lessons for students.

Khan and his son talked through a math problem with GPT-4o; Khan told the chatbot not to solve the problem for his son, but rather to “nudge him in the right direction.”

ChatGPT-4o can tell bedtime stories and “sing”

During a demo at its unveiling, ChatGPT-4o was asked by OpenAI’s head of frontiers research, Mark Chen, to tell “a bedtime story about robots and love .”

GPT-4o started with: “Once upon a time, in a world not too different from ours, there was a robot named Byte. Byte was a curious robot. Always exploring.” However, Chen said he wanted “a little bit more emotion in your voice, a little bit more drama,” and continued to push the chatbot to have more expressiveness in its voice.

OpenAI chief technology officer Mira Murati then asked GPT-4o to use a robot voice to tell a story, which it did. It also “sang” in its synthetic voice after being asked to by OpenAI researcher Barret Zoph.

ChatGPT-4o can do live translations

OpenAI chief technology officer, Mira Murati, and head of frontiers research, Mark Chen, demonstrated how ChatGPT-4o can do live translations using Murati’s Italian skills.

Chen asked GPT-4o to act as a translator, translating to Italian when it hears English, and translating to English when it hears Italian.

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The Government Takes On Ticketmaster

Explaining a case that could reshape the multibillion-dollar live entertainment industry..

Hosted by Sabrina Tavernise

Featuring David McCabe

Produced by Will Reid ,  Rob Szypko and Rachelle Bonja

Edited by Brendan Klinkenberg and Michael Benoist

Original music by Marion Lozano ,  Dan Powell and Will Reid

Engineered by Alyssa Moxley

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Over recent years, few companies have provoked more anger among music fans than Ticketmaster. Last week, the Department of Justice announced it was taking the business to court.

David McCabe, who covers technology policy for The Times, explains how the case could reshape America’s multibillion-dollar live music industry.

On today’s episode

David McCabe , a technology policy correspondent for The New York Times.

Background reading

The government is accusing Ticketmaster’s corporate parent, Live Nation Entertainment, of violating antitrust laws .

Here’s a guide to the emails at the heart of the government’s case .

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

David McCabe covers tech policy. He joined The Times from Axios in 2019. More about David McCabe

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