Pythagoras Theorem - Formula, Proof, Examples

TL;DR

Pythagoras theorem says a² + b² = c². Learn the formula, four proofs, common mistakes, the 4,000-year-old Babylonian tablet, and worked examples.

Last updated on June 9, 2026 8 min read

The Pythagoras theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides — written algebraically as a² + b² = c², where c is the hypotenuse and a and b are the two legs. The theorem holds for every right triangle, has at least 370 distinct proofs, and shows up in everything from carpentry to GPS calculation. The Greeks formalised it; the Babylonians, Egyptians, Chinese, and Indians had been using it for centuries before Pythagoras was born.

Quick Reference

What is the Pythagoras Theorem?

In any right-angled triangle, the side opposite the right angle is the longest — call it the hypotenuse. The Pythagoras theorem states that if you square the lengths of the two shorter sides and add them, the sum equals the square of the hypotenuse.

So if a right triangle has legs of length 3 and 4, the hypotenuse is \sqrt{3^2 + 4^2} = \sqrt{25} = 5. The triple (3,4,5) is the most famous example of a Pythagorean triple — three whole numbers that satisfy the theorem. Other common triples are (5,12,13), (8,15,17), (7,24,25).

The converse is also true: if three sides a, b, c of a triangle satisfy a² + b² = c², the triangle is necessarily right-angled at the corner opposite c. Builders use this fact daily — the 3-4-5 rule for setting square corners is the converse in action.

Four Ways To Prove The Pythagoras Theorem

Proof 1: Visual rearrangement. Take four identical right triangles with legs a, b and hypotenuse c. Arrange them inside a square of side a + b in two ways.

The first arrangement leaves an empty square of side c in the middle. The second arrangement leaves two empty squares of sides a and b. Both empty regions have the same total area, because the four triangles take up the same area both times. Therefore c² = a² + b².

Proof 2 (algebraic, using the same arrangement). The big square has area (a+b)² = a² + 2ab + b². The four triangles have total area 4 \cdot \frac{1}{2}ab = 2ab. The middle square has area c².

So c² + 2ab = a² + 2ab + b², which simplifies to a² + b² = c².

Proof 3 (similar triangles). Drop a perpendicular from the right-angle vertex to the hypotenuse. This splits the original triangle into two smaller triangles, both similar to the original. Comparing ratios from those similarities gives the result directly.

Proof 4 (Garfield's proof). US President James A. Garfield published a proof using a trapezium of two right triangles. The trapezium's area, computed two ways, again yields a² + b² = c².

Worked Examples of Pythagoras Theorem

Example 1: Find the hypotenuse

A ladder leans against a wall. The foot of the ladder is 5 m from the wall, and the top reaches 12 m up. How long is the ladder?

The ladder forms the hypotenuse of a right triangle with legs 5 and 12.

c² = 5² + 12² = 25 + 144 = 169

c = \sqrt{169} = 13

Final answer: The ladder is 13 m long.

Example 2: Find a missing leg (the wrong path first)

A right triangle has hypotenuse 25 and one leg 7. Find the other leg.

The instinct is to write a² + b² = c² as 7² + b² = 25² and grind through. But if we treat the longest given side as a leg:

b² = 625 + 49 = 674, b = \sqrt{674} ≈ 25.96

That cannot be right. The correct setup keeps 25 as the hypotenuse:

7² + b² = 25² 49 + b² = 625

b² = 576

b = 24

Final answer: The other leg is 24.

Example 3: Use the converse to check a right angle

A triangle has sides 9, 40, 41. Is it right-angled?

Check whether 9² + 40² = 41²:

9² + 40² = 81 + 1600 = 1681 41² = 1681

The two are equal, so the triangle is right-angled at the corner opposite the side of length 41.

Final answer: Yes — (9,40,41) is a Pythagorean triple, so the triangle is right-angled.

The Mathematicians Who Shaped The Pythagoras Theorem

The story does not begin with Pythagoras. The Babylonian tablet YBC 7289 (around 1800 BCE) shows a unit square with its diagonal labelled in cuneiform numerals — the diagonal value is 1.41421, accurate to six decimal places of 2√2. Babylonian scribes were calculating with the theorem long before the Greeks.

Pythagoras of Samos (c. 570–495 BCE, Greece) led a philosophical-mathematical school that proved the result with geometric arguments. Bhāskara II (1114–1185, India) gave a one-word proof, drawing the rearrangement diagram and writing simply "Behold!" — the visual proof speaks for itself. Euclid (c. 300 BCE, Greece) gave the most cited rigorous proof, in Elements Book I, Proposition 47.

Common Mistakes of Pythagoras Theorem

Mistake 1: Treating the longest given side as a leg.

Where it slips in: When the problem gives one leg and the hypotenuse, but the student hasn't paused to identify which is which.

The correct way: Identify the hypotenuse first (always opposite the right angle, always the longest side). Then write a² + b² = c² with the hypotenuse on the right.

Mistake 2: Forgetting to take the square root.

Where it slips in: Computing a² + b², getting 169, and writing the answer as 169 instead of 13.

The correct way: Compute a² + b², then take the square root.

Mistake 3: Using the theorem on a non-right triangle.

Where it slips in: The problem doesn't say "right triangle" but the student assumes it is.

The correct way: Confirm the triangle is right-angled.

Mistake 4: Forgetting Pythagorean triples are not the only right triangles.

Where it slips in: The student memorises (3,4,5), (5,12,13), and tries to spot one in every problem.

The correct way: Most right triangles have irrational hypotenuse.

Frequently Asked Questions

Did Pythagoras actually discover the Pythagoras theorem?

No. Babylonian, Egyptian, Chinese, and Indian mathematicians used the relationship for centuries before Pythagoras.

How many proofs of the Pythagoras theorem exist?

Elisha Loomis catalogued 370 in his book The Pythagorean Proposition.

Does the Pythagoras theorem work in three dimensions?

Yes — extended naturally. For a rectangular box with edges a, b, c, the space diagonal has length \sqrt{a² + b² + c²}.

What is a Pythagorean triple?

Three positive integers (a, b, c) that satisfy a² + b² = c². The smallest is (3,4,5). There are infinitely many.

When does the Pythagoras theorem fail?

It fails on non-right triangles and curved surfaces.