Sheet № 107 · Foundation + Higher · AQA · Edexcel · OCR
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.
§Key definitions
Question:
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
Answer:
c = 10 cm. (This is the Pythagorean triple 6-8-10, which is double the 3-4-5 triple.)
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?
§Formulas to memorise
c² = a² + b²
c = √(a² + b²)
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
Square both known sides and add them: c² = a² + b².
Worked example
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
⚠ 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.