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: $a^2 + b^2 = c^2$. This visualization shows squares built on each side of the triangle.

Difficulty: beginner

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 = \sqrt{25} = 5
    

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

Conclusion

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.