Squares, triangles, and one deceptively simple equation that builders, gamers and astronauts all rely on every single day. Here's the whole story — with a live triangle you can stretch and measure.
If you've ever used a screen, hung a picture frame, or measured a ladder, you've already used the Pythagorean theorem — probably without realising it. It's one of the oldest and most useful relationships in all of mathematics, and it connects just three things: the two shorter sides of a right triangle and its longest side.
a² + b² = c²
The square of the hypotenuse equals the sum of the squares of the other two sides.
Right angle only
Works exclusively for right-angled triangles — one angle must be exactly 90°.
Find any missing side
Rearrange to solve for a leg: a = √(c² − b²), or for the hypotenuse: c = √(a² + b²).
What Is the Pythagorean Theorem?
SIMPLE DEFINITIONThe Pythagorean theorem states that in any right-angled triangle, the square of the longest side (called the hypotenuse) is exactly equal to the sum of the squares of the two shorter sides (called the legs).
Visually, it means this: if you drew a square on each of the two shorter sides, their total area would be exactly the same as the area of the square drawn on the hypotenuse. That geometric picture is the heart of the theorem.
The hypotenuse is always opposite the right angle, and it is always the longest side. If you label the longest side as anything other than c, you'll tie yourself in knots when rearranging the formula.
Interactive: Stretch a Right Triangle
Drag the two sliders below to change the length of each leg. The hypotenuse recalculates instantly using c = √(a² + b²). Watch how the squares grow and shrink with each side.
Try 3 and 4 — you'll land exactly on 5, one of the famous Pythagorean triples. Then try 5 and 12 for another.
Rearranging the Formula: Finding Any Side
The theorem isn't only for finding the hypotenuse. You can rearrange it to find a missing leg if you know the other leg and the hypotenuse.
Notice the minus sign. To find a leg, you subtract the known leg's square from the hypotenuse's square, then take the square root.
Worked example: finding the hypotenuse
A right triangle has legs of 6 cm and 8 cm. How long is the hypotenuse?
- Write the formula: c² = a² + b²
- Substitute: c² = 6² + 8² = 36 + 64 = 100
- Take the square root: c = √100 = 10 cm
Worked example: finding a missing leg
A right triangle has a hypotenuse of 13 m and one leg of 5 m. How long is the other leg?
- Rearrange: a² = c² − b²
- Substitute: a² = 13² − 5² = 169 − 25 = 144
- Take the square root: a = √144 = 12 m
Why Does It Work? A Visual Proof
THE SQUARE PROOFImagine a square with sides of length (a + b). Inside it, arrange four identical copies of your right triangle, leaving a tilted square hole in the middle whose side is the hypotenuse, c.
- The area of the big square is (a + b)² = a² + 2ab + b².
- The four triangles together have area 4 × (½ × a × b) = 2ab.
- The tilted hole in the middle has area c².
- So: a² + 2ab + b² = 2ab + c².
- Cancel the 2ab from both sides: a² + b² = c².
No algebra tricks, no hidden assumptions — just areas of shapes and a bit of cancellation. This is one of hundreds of known proofs; Pythagoras' original is lost to history, but this one is a classic classroom favourite.
Pythagorean Triples: The Whole-Number Sets
Some right triangles have all three sides as whole numbers. These sets are called Pythagorean triples, and they're worth memorising — they let you solve problems instantly without a calculator.
| a (leg) | b (leg) | c (hypotenuse) | Verification |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 = 5² |
| 5 | 12 | 13 | 25 + 144 = 169 = 13² |
| 8 | 15 | 17 | 64 + 225 = 289 = 17² |
| 7 | 24 | 25 | 49 + 576 = 625 = 25² |
| 20 | 21 | 29 | 400 + 441 = 841 = 29² |
Any multiple of a triple is also a triple: 6-8-10, 9-12-15, 10-24-26, and so on. Spotting these saves time in exams and real-world calculations alike.
Real-World Uses of the Pythagorean Theorem
This isn't an abstract classroom fact — it's used constantly in jobs and everyday life.
- Construction: Builders use the 3-4-5 triple to check that a corner is perfectly square before pouring concrete or laying tiles.
- Screens and monitors: The advertised diagonal size of a TV or laptop screen is the hypotenuse of its width and height.
- Navigation and GPS: The shortest straight-line distance between two points on a flat map is calculated using the theorem.
- Ladders and ramps: Working out how far up a wall a ladder reaches, given its length and distance from the wall.
- Computer graphics and gaming: Calculating distances between objects on screen — every time a game engine checks if two characters are close enough to collide, Pythagoras is running in the background.
Common Mistakes to Avoid
- Using it on non-right triangles: The theorem only works when one angle is exactly 90°. Otherwise you need the Law of Cosines.
- Forgetting to take the square root: You find c² first — remember that c itself is the square root of that number.
- Mixing up hypotenuse and leg: When finding a missing leg, always subtract from the hypotenuse's square, never add.
- Rounding too early: Keep full precision until the final step to avoid small errors piling up.
A Brief History
Pythagoras was a Greek philosopher who lived around 570–495 BCE, and the theorem carries his name in the West. But Babylonian clay tablets show that the 3-4-5 relationship was known over a thousand years earlier. What Pythagoras (or his school) likely contributed was the first formal proof — the step from "it works for these numbers" to "it works for every right triangle, forever."