Pythagoras' Theorem (Years 8–11) — Maths

Pythagoras' theorem is a rule for right-angled triangles — triangles with a perfect 90° corner. It links the three sides so tightly that if you know any two, you can always find the third. Builders use it to check a corner is square; it is one of the oldest and most useful results in maths.

Why this matters: the theorem turns up wherever a right angle does. The diagonal of a TV screen, whether a ladder reaches a window, the shortest path across a park, the "3-4-5" trick builders use to set out a square corner — all are Pythagoras. It is also the foundation the next topic (trigonometry) is built on, so it is well worth owning completely.

a² + b² = c² — c is the longest side, opposite the right angle

a² + b² = c²
where c is the hypotenuse — the longest side, always opposite the right angle.

The two smaller squares together equal the square on the hypotenuse.

The two shorter sides (a and b) are the "legs" meeting at the right angle; c faces away from it. Because c is longest, its square (c²) is bigger than either of the others — a handy sense-check on every answer.

Finding the hypotenuse (the long side)

Add the squares of the legs, then square-root: the classic 3‑4‑5.

Square the two shorter sides, add them together, then square-root:

Legs 3 and 4: c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5.

Finding a shorter side

For a leg, subtract the known square from the hypotenuse's.

Rearrange: square both known sides, subtract the smaller square from the larger (the hypotenuse's), then square-root:

Hypotenuse 13, one leg 5: b² = 13² − 5² = 169 − 25 = 144, so b = √144 = 12.

You want	Operation	Then
Hypotenuse (long side)	a² + b² (ADD)	square-root
A shorter side (leg)	c² − known² (SUBTRACT)	square-root

A few "Pythagorean triples" — whole-number sets that fit exactly — are worth memorising, as they crop up constantly: 3, 4, 5 · 5, 12, 13 · 8, 15, 17 · 7, 24, 25. Any multiple works too, so 6, 8, 10 is just 3, 4, 5 doubled.

Common mistake: adding when you should subtract. Add the squares only when finding the hypotenuse; subtract when finding a shorter side. Quick check: the hypotenuse must always be the biggest number — if your "long side" comes out shorter than a leg, you added by mistake.

Memory hook: a² + b² = c², with c the longest side opposite the right angle. Add to find the hypotenuse, subtract to find a leg, then square-root either way.