# Pythagorean Theorem calculator: Calculating the Third Side of a Triangle

## Introduction

A right triangle is a type of triangle that has one angle measuring 90 degrees. The Pythagorean theorem is a fundamental concept in mathematics that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

## Calculating the Third Side of a Right Triangle

If you know the lengths of any two sides of a right triangle, you can use the Pythagorean theorem calculator to calculate the length of the third side. This can be done using a right triangle sides calculator.

To calculate the length of a side, enter 0 for the side you are looking for.

## The Pythagorean Theorem

To understand the Pythagorean Theorem, let’s consider a right triangle with sides a, b, and c. The side opposite the right angle is called the hypotenuse (c), while the other two sides are known as the legs (a and b).

The Pythagorean theorem can be written as:

a² + b² = c²

Where:

- a and b are the lengths of the two legs of the right triangle
- c is the length of the hypotenuse

### Calculating the Hypotenuse

If you know the lengths of both legs of a right triangle, you can use the Pythagorean theorem calculator to calculate the length of the hypotenuse. Simply square the lengths of the two legs, add them together, and then take the square root of the sum. The formula for calculating the hypotenuse is:

c = √(a² + b²)

where:

- c is the length of the hypotenuse
- a and b are the lengths of the legs

### Calculating a Leg

If you know the length of the hypotenuse and one leg of a right triangle, you can use the Pythagorean theorem to calculate the length of the other leg. Subtract the square of the known leg from the square of the hypotenuse, and then take the square root of the difference. The formula for calculating a leg is:

a = √(c² – b²)

or

b = √(c² – a²)

where:

- a and b are the lengths of the legs
- c is the length of the hypotenuse

## Proof of the Pythagorean Theorem

There are several ways to prove the Pythagorean Theorem, but one of the most well-known proofs is attributed to the ancient Greek mathematician Pythagoras.

Pythagoras’ proof involves using squares to represent the areas of the three sides of a right triangle. By rearranging these squares, he demonstrates that the sum of the areas of the squares on the two smaller sides is equal to the area of the square on the hypotenuse.

This proof provides a visual representation of why the Pythagorean Theorem holds true and helps to deepen our understanding of the concept.

## Using the Pythagorean Theorem calculator

Calculating the sides of a right triangle manually can be time-consuming and prone to errors. To simplify the process, you can use a right triangle sides calculator. These calculators are available online and allow you to input the known side lengths and calculate the unknown side length automatically.

Here’s how you can use a right triangle sides calculator:

- Enter the known side lengths into the appropriate fields. For example, if you know the lengths of the two legs, enter them into the corresponding fields.
- Select the option to calculate the unknown side length. If you want to find the hypotenuse, choose the hypotenuse calculator option. If you want to find one of the legs, choose the leg calculator option.
- Click on the “Calculate” or “Calculate Now” button to perform the calculation.
- The calculator will display the length of the unknown side.

## Conclusion

The Pythagorean theorem is a powerful tool for calculating the lengths of the sides of a right triangle. By using a calculator for the hypotenuses and legs of a triangle, you can quickly and accurately determine the length of the hypotenuse or one of the legs. Whether you’re a student studying geometry or a professional working with triangles, a right triangle sides calculator can save you time and effort in your calculations.

Remember, the Pythagorean theorem applies only to right triangles, so make sure you have the correct type of figure before using the calculator.