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Applying Pythagoras
Rung 2 of 4 · The method

Both Cases, Step by Step

Two flavours of the same job. Finding the hypotenuse: add, then root. Finding a shorter side: subtract, then root. Let's drill both until they're reflex.


PractiseFlip between the two cases, hit “new triangle”, work it out, then check yourself — or reveal the steps.
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Audio WalkthroughComing Soon
Video ExplainerComing Soon

Every Pythagoras question is one of two cases, and you tell them apart by asking a single question: is the side I want the hypotenuse, or a shorter side? That answer decides whether you add or subtract.

Case One — Finding the Hypotenuse

You've got both legs and want the long slanted side. Square each leg, add them, square-root. Worked example: legs 5 and 12 → 5² + 12² = 25 + 144 = 169, then c = √169 = 13. Tidy, because 5-12-13 is a perfect set; most won't be, so you round to one decimal place — like √52 ≈ 7.2.

Case Two — Finding a Shorter Side

Now you know the hypotenuse and one leg, and want the other leg. You subtract: take the known leg's square away from the hypotenuse's square, then square-root. Worked example: hypotenuse 13, one leg 5 → a² = 13² − 5² = 169 − 25 = 144, then a = √144 = 12. Same theorem, rearranged.

Say it plainly: hypotenuse? add the squares then root. Shorter side? subtract the squares then root. Either way, the square root is what turns the area back into a length — and you round to 1 decimal place.

One Method, Two Doors

It helps to write a² + b² = c² first every time, then plug in what you know and see which letter is missing. If c is missing you'll be adding; if a leg is missing you'll be subtracting. The toy throws both kinds at you so the “which one is this?” decision becomes instant — which matters, because mixing them up is the whole next rung.

Us, Thinking Out Loud

How do you decide, in one glance, whether to add or subtract?

Why does finding a shorter side use subtraction?