In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². Add the squares to find the hypotenuse; subtract to find a shorter side.
The key idea
Pythagoras’ theorem only works in right-angled triangles.
Always identify the hypotenuse first. It is opposite the right angle and is always the longest side.
Why it works. Draw a square on each side of a right-angled triangle. The two smaller squares always have the same total area as the square on the hypotenuse.
For sides 3, 4 and 5: 3² = 9, 4² = 16, and 9 + 16 = 25 = 5².
a² + b² = c²
Worked examples
Example 1The two shorter sides of a right-angled triangle are 6 cm and 8 cm. Find the hypotenuse.
Finding the hypotenuse: add the squares
- c² = a² + b²Write the formula
- c² = 6² + 8²Put in the two sides you know
- c² = 36 + 64Square each one
- c² = 100Add
- c = √100Square root both sides
- c = 10 cmAnswer, with units
Answer: 10 cm
Example 2The hypotenuse is 13 cm and one side is 5 cm. Find the other side.
Finding a shorter side: subtract the squares
- a² + b² = c²Write the formula
- a² + 5² = 13²The unknown is a shorter side
- a² + 25 = 169Square the known sides
- a² = 169 − 25Subtract to get a² on its own
- a² = 144
- a = √144 = 12 cmSquare root, then units
Answer: 12 cm
Example 3A 5 m ladder leans against a wall with its foot 1.5 m from the wall. How high up the wall does it reach?
A real-life problem: the ladder is the hypotenuse
- h² + 1.5² = 5²The ladder (5 m) is opposite the right angle
- h² + 2.25 = 25Square the known lengths
- h² = 25 − 2.25 = 22.75Subtract, because h is a shorter side
- h = √22.75 = 4.769…Square root
- h ≈ 4.77 mRound sensibly: 3 significant figures
Answer: About 4.77 m
Common mistakes
- Adding the squares when finding a shorter side. If you know the hypotenuse, subtract.
- Forgetting to take the square root at the end.
- Using the theorem in a triangle that isn’t right-angled.
Practice questions
- The shorter sides are 9 cm and 12 cm. Find the hypotenuse.
Show answer
15 cm
- The hypotenuse is 17 cm and one side is 8 cm. Find the other side.
Show answer
15 cm
- The shorter sides are 2 cm and 3 cm. Find the hypotenuse to 2 decimal places.
Show answer
√13 ≈ 3.61 cm
- Is a triangle with sides 7, 24 and 25 right-angled?
Show answer
Yes: 7² + 24² = 49 + 576 = 625 = 25²
More topic explainers
- Solving linear equations (MYP 2–3)
- Right-angled trigonometry (SOH CAH TOA) (MYP 4)
- Solving quadratic equations (MYP 4–5)
Find more free resources, or look up command terms in the glossary.