The Pythagorean theorem has more proofs than any other result in mathematics — over 370 documented proofs, contributed across 2,500 years by everyone from Euclid to U.S. President James Garfield. Here are three of the cleanest, each using a different mathematical idea.
What the theorem says
For a right triangle with legs of length a and b and hypotenuse c:
a² + b² = c²
This relationship holds in any right triangle, regardless of size or orientation. It's a statement about areas: the squares built on the two legs have a combined area equal to the square built on the hypotenuse.
Proof 1: Area dissection
This is the most visual proof, no algebra required.
Draw a large square with side length (a + b). Inside, place four identical right triangles (legs a and b) with their hypotenuses forming a tilted square in the middle. The four triangles fill the corners; the inner tilted square has side c.
Now compute the big square's area two ways:
- Direct: (a + b)² = a² + 2ab + b²
- By parts: 4 triangles + tilted square = 4 × (½ab) + c² = 2ab + c²
Setting them equal: a² + 2ab + b² = 2ab + c². Subtract 2ab from both sides: a² + b² = c².
This proof is attributed to Bhaskara, a 12th-century Indian mathematician. It works because rearranging the four triangles inside the big square doesn't change their total area — only the shape of what's left over.
Proof 2: Similar triangles
This proof uses the fact that dropping the altitude from the right angle creates two smaller triangles, each similar to the original.
In a right triangle with the right angle at C, drop a perpendicular from C to the hypotenuse AB, hitting it at point D. This creates three triangles: the original (ABC), and two smaller ones (ACD and BCD). All three are similar.
From similar triangles, corresponding sides are in proportion:
- AD / AC = AC / AB → AC² = AD × AB → b² = AD × c
- BD / BC = BC / AB → BC² = BD × AB → a² = BD × c
Adding: a² + b² = (AD + BD) × c = AB × c = c × c = c².
This is essentially Euclid's proof from the Elements, refined over centuries. It's the proof that ties the theorem to the broader geometric machinery of similar triangles.
Proof 3: Garfield's trapezoid
President James A. Garfield discovered this proof in 1876, five years before becoming president. It uses a trapezoid instead of a square, requiring just half the figure of Bhaskara's proof.
Construct a trapezoid with parallel sides of length a and b, and the two non-parallel sides each of length c. The trapezoid is made of two right triangles (legs a and b, hypotenuse c) plus a third triangle in the middle.
Compute the trapezoid's area two ways:
- Trapezoid formula: ½ × (a + b) × (a + b) = ½(a + b)²
- Sum of triangles: 2 × (½ab) + ½c² = ab + ½c²
Setting them equal: ½(a + b)² = ab + ½c². Multiply through by 2: (a + b)² = 2ab + c². Expand: a² + 2ab + b² = 2ab + c². Subtract 2ab: a² + b² = c².
Why so many proofs?
The Pythagorean theorem connects three big areas of math: geometry (right triangles), algebra (squares of lengths), and number theory (Pythagorean triples like 3, 4, 5). Almost any framework you set up reaches the same conclusion. The theorem is so foundational that proving it from many angles has been a tradition since Euclid.
Loomis's The Pythagorean Proposition (1968) catalogues 370 distinct proofs, including ones using calculus, vectors, complex numbers, and even differential equations.
The converse is also true
If a² + b² = c² for a triangle, then the triangle has a right angle opposite c. This is the converse of the Pythagorean theorem, and it's used to verify right angles — in carpentry, in surveying, and in any geometry problem where you don't know the triangle is right.
Practice with the calculator
Our Pythagorean theorem calculator solves for any missing side — hypotenuse from two legs, or one leg from the hypotenuse and the other leg. It also returns the perimeter, area, and the two acute angles. Useful for checking homework or working through Pythagorean triple patterns.