The Pythagorean Theorem: Proof, Examples and Real-World Uses

In a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$$a^2+b^2=c^2$$

A visual proof

Arrange four copies of the triangle inside a square of side $a+b$. The empty space in the middle is a square of side $c$, so $(a+b{)}^2=4\cdot \frac{1}{2}ab+c^2$, which simplifies to $a^2+b^2=c^2$.

Examples

• Legs 3 and 4: $c=\sqrt{9+16}=5$.
• Hypotenuse 13, leg 5: $b=\sqrt{169-25}=12$.

Real-world uses

• Construction: builders check right angles with the 3-4-5 rule.
• Navigation: straight-line distance between two points uses the distance formula, which is the theorem in disguise.
• Screens: TV sizes are diagonal measurements.

Try the Pythagorean Theorem Calculator.