The Pythagorean theorem may have more proofs than any other result in mathematics — one classic collection alone gathers 370 of them, contributed over more than 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:

  1. Direct: (a + b)² = a² + 2ab + b²
  2. 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 rearrangement proof is old, and its origin is uncertain. The 12th-century Indian mathematician Bhaskara II is credited with a close cousin: he placed the same four triangles inside a square of side c, leaving a small square of side (b − a) in the middle (taking b as the longer leg), so c² = 4 × (½ab) + (b − a)² = 2ab + a² − 2ab + b² = a² + b². Both versions work 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.

Euclid uses this argument in the Elements too (in a lemma to Book X, Proposition 33, and in the Book VI generalization, Proposition 31), although his famous proof of Book I, Proposition 47 compares areas instead. 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: the trapezoid is exactly half of the (a + b) square from Proof 1, cut along a diagonal of the tilted inner square.

Construct a right trapezoid with parallel sides of length a and b and height a + b. It is made of two right triangles (legs a and b, hypotenuse c) plus a third triangle in the middle whose two equal sides are those hypotenuses, each of length c. The angle between them is a right angle, because the two acute angles beside it add up to 90°, so the middle triangle's area is ½c².

Compute the trapezoid's area two ways:

  1. Trapezoid formula: ½ × (a + b) × (a + b) = ½(a + b)²
  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.

Elisha Scott Loomis's The Pythagorean Proposition (second edition 1940, reissued in 1968) catalogues 370 proofs in four groups: 109 algebraic, 255 geometric, 4 quaternionic (built on vector operations) and 2 dynamic (built on mass and velocity). Loomis argued that no trigonometric, analytic-geometry or calculus proof is possible, because those subjects already rest on the theorem, a view some later authors have challenged.

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.

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