Finding the Hypotenuse –

Finding the hypotenuse is the most common application of Pythagoras' theorem in GCSE Maths. Once you can identify the longest side and apply the formula confidently, you unlock marks on both Foundation and Higher papers.

What Is Finding the Hypotenuse?

The hypotenuse is the longest side of a right-angled triangle. It is always the side directly opposite the right angle (the 90° angle, usually marked with a small square). Every right-angled triangle has exactly one hypotenuse.

Pythagoras' theorem tells us that the square of the hypotenuse equals the sum of the squares of the other two sides. If the two shorter sides are a and b, and the hypotenuse is c, the relationship is expressed as a single formula. This formula is not provided on the exam formula sheet — you must memorise it.

To find the hypotenuse, you square the two known sides, add them together, and take the square root of the result. The calculation is straightforward, but identifying the hypotenuse correctly is essential — labelling the wrong side leads to an entirely wrong answer.

Key Formulas

Pythagorean Triples

Pythagorean triples are sets of three whole numbers that satisfy Pythagoras' theorem perfectly, with no rounding needed. Recognising them saves time in exams:

• 3, 4, 5 (and multiples: 6-8-10, 9-12-15, 15-20-25)
• 5, 12, 13 (and multiples: 10-24-26)
• 8, 15, 17
• 7, 24, 25

Step-by-Step Method

1. Identify the right angle and label the hypotenuse (c) — the side opposite it.
2. Label the other two sides a and b (order does not matter).
3. Square both known sides and add them: c² = a² + b².
4. Take the square root of the result to find c.
5. Round your answer as the question instructs (1 d.p., 3 s.f., etc.).

Worked Example 1 — Foundation Level

Question: A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.

Working: c² = a² + b² c² = 6² + 8² c² = 36 + 64 c² = 100 c = √100

Answer: c = 10 cm. (This is the Pythagorean triple 6-8-10, which is double the 3-4-5 triple.)

Worked Example 2 — Higher Level

Question: A right-angled triangle has legs of 7.5 cm and 10 cm. Find the hypotenuse. Give your answer to 1 decimal place.

Working: c² = 7.5² + 10² c² = 56.25 + 100 c² = 156.25 c = √156.25

Answer: c = 12.5 cm. (This is exact — no rounding needed.)

Worked Example 3 — Exam Style

Question: A rectangular field measures 40 m by 30 m. A path runs diagonally from one corner to the opposite corner. How long is the path?

Working: The diagonal of a rectangle creates two right-angled triangles. The diagonal is the hypotenuse. c² = 40² + 30² c² = 1600 + 900 c² = 2500 c = √2500

Answer: The path is 50 m long. (This is the 3-4-5 triple scaled by 10.)

Common Mistakes

• Labelling the wrong side as the hypotenuse. The hypotenuse is always opposite the right angle and always the longest side. If your answer is shorter than one of the given sides, you have made an error.
• Forgetting to square root at the end. The formula gives c², not c. You must take the square root as the final step.
• Subtracting instead of adding. When finding the hypotenuse you always add the squares. Subtraction is only used when finding a shorter side.

Exam Tips

• Draw and label the triangle if one is not provided — mark the right angle and label the hypotenuse.
• Show every step: formula, substitution, addition, then square root. Each step can earn a method mark.
• Check your answer makes sense — the hypotenuse must be longer than either of the other two sides.

Practice Questions

Q1 (Foundation): A right-angled triangle has sides of 9 cm and 12 cm. Find the hypotenuse.

Q2 (Foundation): A right-angled triangle has sides of 5 cm and 7 cm. Find the hypotenuse to 1 decimal place.

Q3 (Higher): A ship sails 24 km east and then 10 km north. How far is the ship from its starting point?

Practise finding the hypotenuse questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Pythagoras' Theorem
• Finding a Shorter Side (Pythagoras)
• Trigonometry SOHCAHTOA

Summary

• The hypotenuse is the longest side of a right-angled triangle, always opposite the right angle.
• Use c² = a² + b² to find it: square both shorter sides, add, then square root.
• Pythagorean triples (3-4-5, 5-12-13, 8-15-17) give whole-number answers and appear frequently in exams.
• Always check that your hypotenuse is longer than both other sides — if it is not, you have an error.