Pythagoras' Theorem – The Rule Behind Every Right Triangle

Pythagoras' theorem is one of the most famous results in all of mathematics: in any right-angled triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. It connects algebra and geometry in a single, elegant equation and underpins everything from construction to computer graphics.

The theorem is named after the Greek philosopher Pythagoras (c. 570–495 BCE), but the relationship was known long before him: a Babylonian clay tablet called Plimpton 322, dated to around 1800 BCE, already lists whole-number side lengths that satisfy the rule – over a thousand years before Pythagoras is credited with proving it in general. What Pythagoras and his followers likely contributed was not the discovery of the pattern but the first general proof that it holds for every right triangle, not just the specific cases the Babylonians had tabulated. Today, more than 370 distinct proofs of the theorem are known, including one published in 1876 by future U.S. President James A. Garfield, discovered while he was still a congressman.

The Theorem

For a right triangle with legs a and b, and hypotenuse c (the side opposite the right angle):
a² + b² = c²

A Simple Proof (Rearrangement)

Take four identical copies of a right triangle with legs a and b, and hypotenuse c. Arrange them inside a square of side (a + b) in two different ways:
1. Arranged around the edges, they leave a tilted square of side c in the middle – its area is c².
2. Rearranged into two rectangles, they leave two smaller squares in the corners – areas a² and b².
Since both arrangements use the same four triangles inside the same big square, the leftover area must be equal either way: c² = a² + b². This is essentially the proof Bhāskara, a 12th-century Indian mathematician, presented with the single word “Behold!”

Worked Examples

Find the hypotenuse of a right triangle with legs 6 cm and 8 cm.

c² = a² + b² = 6² + 8² = 36 + 64 = 100.
c = √100 = 10 cm.

A right triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg.

b² = c² − a² = 17² − 8² = 289 − 64 = 225.
b = √225 = 15 cm.

The Converse of Pythagoras' Theorem

The converse lets you go the other way: if the three sides of a triangle satisfy a² + b² = c² (where c is the longest side), then the triangle must contain a right angle. This is exactly how ancient Egyptian surveyors are believed to have marked out perfect right angles using a simple 3-4-5 knotted rope, long before anyone had proved why it worked.

A triangle has sides 9 cm, 12 cm, and 15 cm. Is it a right triangle?

Longest side is 15 cm, so test: 9² + 12² = 81 + 144 = 225.
15² = 225.
Since 225 = 225, the equation holds – this is a right triangle.

A triangle has sides 5 cm, 6 cm, and 8 cm. Is it a right triangle?

Longest side is 8 cm, so test: 5² + 6² = 25 + 36 = 61.
8² = 64.
Since 61 ≠ 64, the equation does not hold – this is not a right triangle.

Pythagorean Triples

A Pythagorean triple is a set of three positive whole numbers (a, b, c) that satisfy a² + b² = c². The most famous is (3, 4, 5). Scaling any triple by a whole number gives another valid triple – for example (6, 8, 10) and (9, 12, 15) are both multiples of (3, 4, 5).

Triple	Check
(3, 4, 5)	9 + 16 = 25
(5, 12, 13)	25 + 144 = 169
(8, 15, 17)	64 + 225 = 289
(7, 24, 25)	49 + 576 = 625
(20, 21, 29)	400 + 441 = 841

Real-World Applications

Construction and carpentry – checking that a corner is a true right angle using the 3-4-5 rule.
Navigation – finding the straight-line distance between two points given their east-west and north-south displacements.
Coordinate geometry – the distance formula between two points (x₁, y₁) and (x₂, y₂) is simply Pythagoras' theorem applied to the horizontal and vertical gaps: d = √[(x₂−x₁)² + (y₂−y₁)²].
3D space – the theorem extends to find the diagonal of a box: d² = l² + w² + h².
Screens and TVs – a screen's diagonal size is calculated from its width and height using Pythagoras' theorem.

Key Takeaways

Pythagoras' theorem: a² + b² = c², where c is the hypotenuse of a right triangle.
The converse works in reverse: if a² + b² = c², the triangle has a right angle.
Pythagorean triples like (3,4,5) and (5,12,13) give whole-number side lengths and can be scaled up.
The theorem extends beyond triangles: it underlies the distance formula in coordinate geometry and 3D space.

Practice: Pythagoras' Theorem

Finding the Hypotenuse