In any right-angled triangle, a2 + b2 = c2. Here's how to use it to find a missing side — and the one thing you must check first.
Pythagoras' theorem is one of the most useful results in all of maths. It says that in a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.
Here c is the hypotenuse — the side opposite the right angle, and always the longest. a and b are the two shorter sides.
1Finding the hypotenuse
- 1Write the theorem: c2 = a2 + b2.You're looking for the longest side, so c is the unknown.
- 2Substitute: c2 = 32 + 42 = 9 + 16 = 25.Square each short side and add.
- 3Square-root both sides: c = 25 = 5.Undo the square to get the length.
2Finding a shorter side
If the hypotenuse is known and you want a shorter side, rearrange: a2 = c2 − b2. Subtract instead of add.
Only add when finding the hypotenuse. To find a shorter side you must subtract the squares. Mixing this up gives an impossible triangle (or a maths error).
Pythagoras only works in a right-angled triangle. Always check there's a 90° angle before using it.
This theorem returns in Grade 10–12 analytical geometry (the distance formula is Pythagoras in disguise) and trigonometry, so it's well worth mastering now.
Practice exercises
Work each one out, then click to reveal the answer.
- 1A right triangle has legs 6 and 8. Find the hypotenuse.
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c2 = 36 + 64 = 100 → c = 10 - 2Legs of 5 and 12 — find the hypotenuse.
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c2 = 25 + 144 = 169 → c = 13 - 3Hypotenuse 10, one leg 6. Find the other leg.
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b2 = 100 − 36 = 64 → b = 8 - 4Is a triangle with sides 3, 4, 6 right-angled?
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32 + 42 = 25 but 62 = 36 — not equal, so no
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