# What Is Pythagoras' Theorem?

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## What Is Pythagoras' Theorem? (The Lesson)

Pythagoras’ theorem is a theorem concerning the three sides of a right triangle.

Pythagoras' theorem (or the Pythagorean theorem) states that:

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

It is easier to remember Pythagoras' theorem as a formula:

In the formula, **c** is the length of the hypotenuse (the longest side, opposite the right angle) and **a** and **b** are the lengths of the other two, shorter sides. The image below shows what we mean:

## Interactive Widget

Use this **interactive widget** to create a right-angled triangle and then use Pythagoras' Theory to calculate the hidden side. Start by selecting which side you want to hide (using the green buttons) and then clicking in the shaded area.

Hypotenuse: ? | |

Opposite: 0 | |

Adjacent: 0 |

**Oops, it's broken!Turn your phone on its side to use this widget.**

## Visualizing Pythagoras' Theorem

The square of a side of a right triangle can be visualised by drawing a square on that side.

Pythagoras' theorem says that the area of the square on the longest side is equal to the sum of the areas of the squares on the other two sides.

By Pythagoras' theorem, the area of the red square is equal to the areas of the blue and green squares added together.

## A Real Example of Pythagoras' Theorem

### Question

The image below shows a right triangle with sides of 3, 4 and 5.

Show that Pythagoras' theorem works for this right triangle.

### Step-by-Step:

# 1

Start with the formula for Pythagoras' theorem:

a^{2} + b^{2} = c^{2}

**Don't forget:** a^{2} = a × a (a squared) **and** b^{2} = b × b **and** c^{2} = c × c.

# 2

Find the lengths of the sides from the right triangle.

In our example, the two shorter side lengths are a = 3 and b = 4. The longest side length is c = 5.

# 3

Substitute a = 3, b = 4 and c = 5 into the formula.

In our example, the two shorter side lengths are a = 3 and b = 4. The longest side length is c = 5.

3^{2} + 4^{2} = 5^{2}

(3 × 3) + (4 × 4) = 5 × 5

9 + 16 = 25

### Answer:

9 + 16 does equal 25. Pythagoras' theorem works.

**Did you know?:** a 3,4,5 triangle is known as a Pythagorean triple. All three sides are integers (see **Note**).