Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². This visualization shows squares built on each side of the triangle.

Difficulty: beginner

Notes

The Pythagorean Theorem is one of the most important ideas in geometry. It describes the relationship between the sides of a right triangle — a triangle with one right angle (90°).

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

Written as a formula:

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

where $a$ and $b$ are the two shorter sides (called legs) and $c$ is the hypotenuse — the side opposite the right angle.

Understanding the Idea

Imagine a right triangle with a square built on each of its sides. If you find the area of each square:

The big square on side $c$ has area $c^{2}$
The two smaller squares have areas $a^{2}$ and $b^{2}$

The theorem says that the area of the large square equals the combined area of the two smaller ones.

Example: The 3-4-5 Triangle

Suppose a right triangle has legs $a = 3$ and $b = 4$ . Then:

c^{2} = 3^{2} + 4^{2} = 9 + 16 = 25

c = 25 = 5

This is known as a 3-4-5 triangle, one of the most famous Pythagorean triples.

Summary

The Pythagorean Theorem connects the three sides of a right triangle:

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

Why It Matters

The theorem is used in many real-life situations:

Finding the diagonal of a rectangle or square
Measuring the distance between two points
Designing buildings, bridges, and ramps
Calculating paths in computer graphics and engineering

Historical Note

The theorem is named after Pythagoras, a Greek mathematician who lived more than 2,500 years ago. However, similar ideas were known even earlier in ancient Egypt, Babylon, and India.