Pythagorean Theorem Explained Simply (With Examples)

Two years ago I watched a Year 9 lad in a Salford classroom get every Pythagoras question right on a worksheet. Perfect layout, perfect square roots. Then I asked him why we add the squares, and he went quiet. He knew the procedure and had no idea what it was doing. The next test came with a ladder-against-a-wall problem instead of a bare triangle, and the procedure fell over.

The Pythagorean theorem says that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c², where c is the side opposite the right angle. You use it to find a missing length, and its converse to prove that a corner is a right angle.

The idea behind the formula: squares, not lengths

The word that matters in the theorem is "square". Picture a right-angled triangle and build a real square on each of its three sides, each square as wide as the side it sits on. The square on the hypotenuse has exactly the same area as the two other squares put together. That is the whole theorem. a² is an area you could shade in with a pencil, not an abstract number.

Draw it once on squared paper, using the 3-4-5 triangle: a square of 9 little boxes, one of 16, one of 25. 9 + 16 = 25. You can count it, and the formula stops being a spell you recite.

A bit of history, since someone in every class asks. Pythagoras probably didn't discover this. A Babylonian clay tablet called Plimpton 322 (around 1800 BC, kept at Columbia University) lists number triples that satisfy the relationship. The oldest surviving written proof is Euclid's, in the Elements, Book I, Proposition 47.

Three examples, from easiest to most useful

Example 1: finding the hypotenuse

A right-angled triangle has shorter sides of 6 cm and 8 cm. Find c.

c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.

Looking for the hypotenuse means adding. And 10 is bigger than both 6 and 8, which is what we expect.

Example 2: finding a shorter side

The hypotenuse is 13 cm and one shorter side is 5 cm. Find the other side, b.

b² = 13² − 5² = 169 − 25 = 144, so b = √144 = 12 cm.

This time we subtract, because the hypotenuse carries the biggest square and the two others have to add up to it. Mixing this up is the most common error, and I'll come back to it below.

Example 3: the ladder and the baseball diamond

A 5 m ladder leans against a wall with its foot 1.5 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse, opposite the right angle between ground and wall. So h² = 5² − 1.5² = 25 − 2.25 = 22.75, and h = √22.75 ≈ 4.77 m.

The "≈" matters. √22.75 is not a tidy number, so say you've rounded and give the unit.

For the American readers: a baseball diamond has 90 ft between bases, and the throw from home plate to second base is the diagonal of a square. d² = 90² + 90² = 16,200, so d ≈ 127.3 ft. Same theorem, different sport.

The mistake that costs the most marks

Most Pythagoras errors I saw while marking Edexcel scripts came from one reflex: adding the squares every single time. A student looking for a shorter side adds anyway, gets 27.25 for the ladder, takes the root and announces that the ladder reaches 5.22 m up a wall. It's longer than the ladder. Nobody blinks.

A quick sanity check saves you: the hypotenuse is always the longest side. If your answer breaks that rule, you've got a sign wrong somewhere.

Two smaller traps. Forgetting the square root at the end (writing 100 instead of 10, which is an area). And writing (a + b)² when you mean a² + b², which are very different things.

A mum got in touch through EduBoost last spring about her 8th-grade son in Ohio. He "knew" the theorem but kept dropping a point on every quiz. When we went through his papers together, the formula was fine and the picture wasn't: he never marked the hypotenuse before starting. One habit fixed it, a quick circle round the side opposite the right angle before touching a calculator. Two quizzes later he had all his points back.

The converse: proving a corner is a right angle

Flip it round. You're given three lengths and want to know whether the triangle is right-angled. Find the longest side, square it, then add the squares of the other two and compare.

Take 7, 24 and 25. 25² = 625 and 7² + 24² = 49 + 576 = 625. Equal, so the triangle is right-angled. With 4, 5 and 7: 4² + 5² = 41, but 7² = 49. Not equal, so no right angle.

That's where the site-foreman trick comes from. A 3-4-5 triangle made from string gives a perfect right angle without a set square.

Five-question quiz

1. A right-angled triangle has shorter sides of 9 cm and 12 cm. How long is the hypotenuse?
2. The hypotenuse is 10 cm and one shorter side is 6 cm. How long is the other side?
3. Is a triangle with sides 5, 12 and 13 right-angled?
4. Is a triangle with sides 4, 5 and 7 right-angled?
5. A pizza box measures 12 inches by 16 inches. What is the diagonal of the lid?

Answers: 1) 15 cm (81 + 144 = 225). 2) 8 cm (100 − 36 = 64). 3) Yes (25 + 144 = 169 = 13²). 4) No (16 + 25 = 41, not 49). 5) 20 inches (144 + 256 = 400).

If you miss two or more, try the Feynman technique: explain out loud, in plain words, why you subtract in Example 2. The gap shows up immediately. To make it stick until the exam, spaced repetition beats a big cram: one Pythagoras question every three or four days does more than ten the night before. Our full guide to spacing your revision has the schedules, and the GCSE maths key topic exercises show where Pythagoras turns up in Higher and Foundation papers.

For practice at your own pace, the AI tutor generates harder variations and goes back over each mistake. Families in England can start with Year 9 maths tutoring or the GCSE maths hub, and families in the US with 8th-grade math tutoring. If you'd rather work with a person, look at our online tutoring options.

Pythagoras boils down to one question before every calculation: where is the right angle? The rest follows. So what trips up your child more, the formula or the diagram?