# Pythagorean Theorem Word Problems Worksheets with Answers

The Pythagorean theorem is one of the most recognizable formulas in mathematics. Even many non-mathematics students recall that the sum of the squares of the two sides in a right triangle are equal to the square of the hypotenuse.

And while this formula may be considered basic after being introduced in the 8th grade, practicing applying it to solve math problems in the real-world is still very important! That’s why I am excited to shared this collection of Pythagorean theorem word problems worksheets with answers!

## What is Pythagorean Theorem Used For?

Pythagoras’ theorem is an example of a mathematical equation that is used to solve math problems involving right triangles. More specifically, we use the Pythagorean theorem to find the value of an unknown side length in a right triangle.

We can see the use of the Pythagorean theorem in many careers and real-world situations in everyday life. For example, carpenters, architects, and navigators apply this formula in order to find the length of the hypotenuse of a triangle found in their professions.

Whether it is a finding where to place the base of the ladder to reach a high window, or to find the total feet of the string needed to complete a triangular design, the Pythagorean theorem has a place in most sample questions you might face when working with real-world triangle problems.

## How to Use the Pythagorean Theorem

The first and most important thing I always start with when explaining Pythagoras’ theorem to my students is that it only holds true for *right triangles.* One of the most common mistakes that I see is students applying the formula when working with non-right triangles.

The Pythagorean theorem is an example of an algebraic formula. In order to understand how to use the Pythagorean Theorem, we first need to take a look at this famous formula:

$$a^2+b^2=c^2$$

In this formula:

*a*represents the length of one of the straight lines that makes up a side length in a triangle*b*represents the length of the second side length*c*represents the length of the hypotenuse of a triangle

Note that *a* and* b* can be used interchangeably without impacting your calculation. However, it is important that the hypotenuse takes the place of *c* in the formula above.

Remember that the hypotenuse is the side length across from the right angle. Since the right angle is the biggest angle in a right triangle, it makes sense that the hypotenuse is also the longest side!

In words, this formula tells us that the sum of the squares of the two sides is equal to the square of the hypotenuse.

An alternate equation for the Pythagorean theorem has the length of the hypotenuse isolated:

$$c=\sqrt{a^2+b^2}$$

In words, this formula tells us that the length of the hypotenuse is equal to the square root of the sum of the squares of the two sides. This formula can be used as a quicker way of solving for the length of the hypotenuse of a triangle.

When working with math problems involving a missing side in a right triangle, we simply substitute the length of each side into either formula for either *a, b, *or *c. *

Let’s take a look at an example!

## Pythagorean Theorem Word Problem Example

*A ship’s mast stands 40 metres tall in the middle of a ship. To ensure the mast’s stability, the captain decides to reinforce it with a support cable that will be attached from the top of the mast to a point on the ground that is 25 metres from the base of the mast. How long should the cable be?*

Since the cable connecting the top of the mast to the point from the base of the mast forms a right triangle, we know that we can apply the Pythagorean theorem to solve for the length of the cable. Your starting point should be to draw a diagram in order to help you identify and label the sides of the right triangle.

Next, we substitute the information from the problem into the formula:

$$25^2+40^2=c^2$$

Simplifying the left side and taking the square root of both sides will allow us to solve for the missing side:

$$\begin{split}25^2+40^2&=c^2 \\ \\ 2225&=c^2 \\ \\ c&=47.17 \end{split} $$

We could also solve this problem using the alternate formula shown above:

$$\begin{split} c&=\sqrt{25^2+40^2} \\ \\ c&=47.17 \end{split} $$

Therefore, the length of the cable will be approximately 47.17 feet.

## Pythagorean Theorem Word Problems Worksheets with Answers

This first worksheet will focus on basic math problems involving right triangles found in everyday life. To solve these problems, keep the equation for the Pythagorean theorem handy. Your starting point should always be drawing a sketch of the right triangle to help you identify the side of the right triangle. Remember to check the answer key to ensure you fully understand!

**Download the PDF worksheet by clicking below!**

This second Pythagorean theorem word problems worksheet with answers will focus on more challenging math problems involving right triangles. The sample questions found on this worksheet involve problems where the hypotenuse is given, and the missing side is one of the other two side lengths.

Since you are working with 2-step equations, you will have to apply algebra to rearrange the equation in order to solve for the measurement of the unknown side length. You can watch this short video example of a problem like this for more help! Be sure to check the answer key if you are feeling stuck!

**Download the PDF worksheet by clicking below!**

## Using the Pythagorean Theorem to Solve Word Problems

Like any other real-life word problems, the first step you always want to take is drawing a diagram. This is a key step that helps you make decisions about the side lengths you are given and the missing side length in the problem. After you have done this, it becomes much easier to determine what value to substitute in each position of the Pythagorean theorem equation!

As is the case with any math concept, the best thing you can do to learn how to apply the Pythagorean theorem is practice! Use the Pythagorean theorem word problems worksheets with answers I have provided here to help you practice solving the various math problems you can encounter involving right triangles!

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