# 20 Pythagorean Theorem Worksheet with Answers (Word & PDF)

If you are familiar with the Pythagorean Theorem, then you know that it is one of the simplest and most useful tools in geometry. The theorem was formulated by Pythagoras of Samos, a Greek mathematician, and philosopher, in the 6th century. Today, it has many applications in different areas of study, from simple elementary and high school geometry to complex engineering and architecture calculations. Below, we take a closer look at the theorem and how you can master it through a worksheet.

## What Is a Pythagorean Theorem Worksheet?

Simply put, the Pythagorean Theorem states that the square of the longest side of a right angle (the hypotenuse) is equal to the sum of the squares of the other two sides. A Pythagorean Theorem Worksheet is a workbook containing different triangle orientations requiring a student to fill in the missing values using the Pythagorean Theorem. It is a great study tool for geometry students.

## Understanding How Pythagoras of Samos Came Up with the Pythagorean Equation

Pythagoras of Samos was a Greek mathematician and philosopher of the 6th century who is widely credited with bringing the Pythagorean Theorem to the attention of the public. While there are other people who used the Pythagorean equation earlier than Pythagoras – most notably the Egyptians – he was the first person to make the connection between the lengths of a right angle triangle’s sides public. This explains why he is widely recognized as the originator and inventor of the equation and why it is named after him – the Pythagorean Equation.

Pythagoras of Samos shared information about the Pythagorean equation with the world in the late 6th century when he resided in Samos. However, many historians maintain that he first made the discovery during his visit to Egypt, where historians believe he learned several branches of mathematics, including Phoenicians arithmetic and geometry.

According to Historians, Pythagoras was very secretive about his work. There is very limited evidence that he even worked on the Theorem or proved it, and credit is often given to his students. Additionally, Pythagoras of Samos was only credited with proving the Theorem over 5 centuries after his demise – the fact that fuels a lot of debate on whether he was behind the theorem at all.

That said, however, Pythagoras of Samos is the most popular figure connected to the Pythagorean Theorem as it is known today. Therefore, it is only fair to give him credit for this amazing equation that has grown to have numerous real-life applications.

## Understanding Pythagorean Theorem

As mentioned, the Pythagorean Theorem states that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. The theorem basically says that if you make squares on each side of a triangle with a 90° angle, the two smaller squares put together will be the same size as the largest square.

The Pythagorean Theorem can be represented mathematically as follows:

a² + b² = c²

Here, c represents the length of the hypotenuse (the longest side), while b and a are the lengths of the other two sides. From the equation, you can easily find the value of one side if you have the values of the other two. Here is an example to demonstrate:

Q. Consider a right-angled triangle with a base of 3cm and a height of 4cm. What is the hypotenuse of the triangle?

A: Using the Pythagorean Theorem to solve, we can assign the base ‘a’ and the height ‘b.’ Substituting in the equation, we have:

a² + b² = c²

3² + 4² = c²

9 + 16 = c²

25 = c²

c = √25

c = 5

The hypotenuse of the triangle is 5 cm long.

As you can see, the hypotenuse was missing from the equation, and we used the relation to find its value. Such questions can be framed differently so that you either have to calculate the value of the base or height. All you have to do is use the Pythagorean Theorem to substitute the missing value. Additionally, you can use the equation to determine whether a triangle is a right triangle.

## How to Use Pythagorean Theorem Worksheet

The Pythagorean Theorem Worksheet is an excellent way to expand your understanding of the Pythagorean Theorem. As you use the worksheet, it is important to remember that ‘a’ and ‘b’ represent the shorter lengths on the triangle while c represents the hypotenuse, which is the longest side.

Pythagoras Theorem worksheets present you with different triangle orientations from which you must determine the value of the missing side. This could be the base, height, or hypotenuse. Regardless of the value you are asked to find; you can use the formula (a²+b²=c²) to solve for the answer. Remember, you can rearrange the equation, but it never changes. Several variations you could use include:

a² + b² = c² (standard)

a² = c² – b² (finding the value of a)

b² = c² – a² (finding the value of b)

Because the worksheet is designed to help familiarize you with the Pythagorean Theorem, the practice questions can be arranged differently to create a challenge. Some examples include:

• Questions using a different set of letters, e.g. (q, r, s) or (x, y, z)
• A question where the triangle sides are labeled in a different order.
• Questions where the sides are named as vertices.

You can also use the worksheet to prove the Pythagorean Theorem and determine whether a triangle is right-angled. As you check your answers, you should always remember:

• The hypotenuse is the longest side of the triangle and the side opposite to the 90° angle.
• While it is the longest side of the triangle, its size can never surpass the sum of the squares of the other 2 sides.

## Conclusion

Pythagoras of Samos formulated the Pythagorean Theorem in the 6th century and, since then, it has been applied to several areas of Mathematics and Geometry. Students can use a Pythagorean Theorem Worksheet to understand the theorem and apply it to solving different questions. Depending on the student’s needs, some of these worksheets have answers, but you can set these pages aside and use them to confirm your answers.