The relationship between the three sides of a right-angled triangle — finding the hypotenuse, finding a shorter side, leaving answers in surd form, and testing whether a triangle is right-angled — with a labelled diagram, worked examples and a quiz.
Around 2 500 years ago it was noticed that the three sides of a right-angled triangle are always locked together by one simple relationship. That relationship is still one of the most useful facts in all of mathematics — builders, surveyors and navigators use it every day.
1Naming the sides
The longest side, always opposite the right angle, is called the hypotenuse. The other two sides are simply the shorter sides (or legs).
a2 + b2 = c2
where c is the hypotenuse
The theorem only works in a right-angled triangle. If there is no right angle, a² + b² = c² is simply false.
2Finding the hypotenuse
- 1Write the theorem: c2 = a2 + b2.You are looking for the longest side, so c is the subject.
- 2c2 = 62 + 82 = 36 + 64 = 100.Square each shorter side, then add.
- 3c = 100 = 10 cm.Take the square root to undo the squaring.
3Finding a shorter side
If the hypotenuse is known and one shorter side is missing, you must subtract rather than add.
- 1Write the theorem: 52 + b2 = 132.13 is opposite the right angle, so it is c.
- 225 + b2 = 169.Square the known values.
- 3b2 = 169 − 25 = 144.Subtract to isolate b2 — this is the key difference.
- 4b = 144 = 12 cm.
Decide first: is the missing side the hypotenuse (then ADD the squares) or a shorter side (then SUBTRACT)? Getting this the wrong way round is the most common Pythagoras error.
4Leaving the answer in surd form
Not every answer is a whole number. When the square root is not exact, CAPS asks you to leave it in surd form — that is, as a root sign rather than a rounded decimal.
- 1c2 = 22 + 32 = 4 + 9 = 13.
- 2c = 13 cm.13 is not a perfect square, so 13 is the exact answer.
- 3(≈ 3,6 cm if a decimal is asked for.)Only round if the question says to.
5Testing whether a triangle is right-angled
Work out the square of the longest side, then the sum of the squares of the other two. If they are equal, the triangle is right-angled.
- 1Longest side is 15: 152 = 225.The longest side would be the hypotenuse.
- 2Other two: 92 + 122 = 81 + 144 = 225.
- 3225 = 225, so YES — it is right-angled.The theorem holds, so the triangle must have a right angle.
- 172 = 49.Longest side.
- 242 + 52 = 16 + 25 = 41.
- 341 ≠ 49, so NO — it is not right-angled.Always state the conclusion in words.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Find the hypotenuse: sides 3 cm and 4 cm.
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c2 = 9 + 16 = 25 → 5 cm - 2Find the hypotenuse: sides 9 cm and 12 cm.
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c2 = 81 + 144 = 225 → 15 cm - 3Find the missing side: hypotenuse 10 cm, one side 6 cm.
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b2 = 100 − 36 = 64 → 8 cm - 4Find the missing side: hypotenuse 25 cm, one side 24 cm.
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b2 = 625 − 576 = 49 → 7 cm - 5Find the hypotenuse in surd form: sides 1 cm and 2 cm.
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c2 = 1 + 4 = 5 → 5 cm - 6Find the hypotenuse in surd form: sides 3 cm and 5 cm.
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c2 = 9 + 25 = 34 → 34 cm - 7Is a triangle with sides 5, 12, 13 right-angled?
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132 = 169; 25 + 144 = 169 → Yes - 8Is a triangle with sides 6, 7, 10 right-angled?
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100 vs 36 + 49 = 85 → No - 9A ladder 5 m long rests 3 m from a wall. How high up the wall does it reach?
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h2 = 25 − 9 = 16 → 4 m
Quick Quiz
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