GeoProof — Pythagorean theorem explorer

a b c

a² = + b² = c² =

The Pythagorean Theorem

In any right triangle — a triangle with one 90° angle — the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

a² + b² = c²

a and b are the two legs that form the right angle; c is the hypotenuse, always the longest side.

What the squares show

The colored squares in the diagram are built directly on each side, so their areas are literally a², b², and c². The theorem says the blue and green areas together exactly equal the pink area — but only when the angle at C is exactly 90°.

The converse also holds

If a² + b² = c², the triangle must be right-angled. Drag a handle and watch the center sign: when angle C is obtuse (>90°), a² + b² < c². When angle C is acute (<90°), a² + b² > c². They're equal only at exactly 90°.

A classic example

The simplest whole-number right triangle is 3-4-5: 3² + 4² = 9 + 16 = 25 = 5². Try typing 3, 4, 5 into the boxes below the diagram.

Where it's used

This relationship underlies distance calculations, navigation, construction (squaring corners), computer graphics, and physics.

A bit of history

Named for the Greek mathematician Pythagoras (c. 570–495 BCE), the relationship was actually known earlier to Babylonian, Indian, and Chinese mathematicians — but his school is credited with the first known general proof.