IGCSE Mathematics: Pythagoras' Theorem Practice Questions
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Add the squares to find the hypotenuse, and subtract to find a shorter side.
Topic 6.2 of Cambridge IGCSE Mathematics 0580 appears in geometry, coordinate geometry and mensuration questions as well as on its own. The one decision to get right is whether to add or subtract, which depends entirely on which side is missing. The questions below cover both plus the 3D case.
What you need to know for Pythagoras' Theorem
- The theorema2 + b2 = c2, where c is the hypotenuse, the side opposite the right angle and always the longest.
- Finding the hypotenuseSquare both shorter sides, add them, then take the square root.
- Finding a shorter sideSquare the hypotenuse, subtract the square of the known shorter side, then take the square root.
- Checking your answerThe hypotenuse must be the largest of the three sides. If your answer for a shorter side exceeds the hypotenuse, you have added instead of subtracted.
- 3D PythagorasFor a cuboid with sides a, b and c, the space diagonal is the square root of (a2 + b2 + c2). It follows from applying the theorem twice.
- Pythagorean triplesWhole number sets such as 3, 4, 5 and 5, 12, 13 and 8, 15, 17. Recognising them saves time.
IGCSE Mathematics Pythagoras' Theorem questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
A right-angled triangle has shorter sides of 9 cm and 12 cm. Calculate the length of the hypotenuse.
Show the worked answer
- The hypotenuse is missing, so add the squares of the two shorter sides.
- 92 = 81 and 122 = 144.
- 81 + 144 = 225.
- The hypotenuse is the square root of 225 = 15 cm. This is the 3, 4, 5 triple scaled by 3.
A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Calculate the length of the other shorter side.
Show the worked answer
- A shorter side is missing, so subtract rather than add.
- 132 = 169 and 52 = 25.
- 169 minus 25 = 144.
- The missing side is the square root of 144 = 12 cm. Check: 12 is less than the hypotenuse of 13, as it must be.
An isosceles triangle has a base of 10 cm and two equal sides of 13 cm. Calculate its perpendicular height and hence its area.
Show the worked answer
- Drop a perpendicular from the apex to the base. It bisects the base, creating two right-angled triangles with a base of 5 cm.
- Each right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm, so the height is the missing shorter side.
- 132 minus 52 = 169 minus 25 = 144, so the height is 12 cm.
- Area of a triangle = half x base x height = 0.5 x 10 x 12 = 60 cm2.
A cuboid measures 3 cm by 4 cm by 12 cm. Calculate the length of the diagonal running from one corner to the opposite corner.
Show the worked answer
- First find the diagonal across the base using the 3 cm and 4 cm sides: 32 + 42 = 9 + 16 = 25, so the base diagonal is 5 cm.
- That base diagonal and the 12 cm height form a right-angled triangle whose hypotenuse is the space diagonal.
- 52 + 122 = 25 + 144 = 169.
- The space diagonal is the square root of 169 = 13 cm. In one step this is the square root of (32 + 42 + 122).
Common mistakes in this topic
- Adding when a shorter side is missing.
- Adding the side lengths rather than their squares.
- Forgetting to take the square root at the end.
- Using the full base of an isosceles triangle instead of half of it.
- Applying the theorem to a triangle with no right angle.
Exam tips
- Identify the hypotenuse first. It is opposite the right angle and always the longest side.
- Missing hypotenuse means add. Missing shorter side means subtract. That single rule covers every 2D question.
- Sense check: no side can be longer than the hypotenuse.
- Learn the common triples. They appear constantly and let you spot the answer instantly.
- For 3D problems, draw the two right-angled triangles separately before calculating.
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Pythagoras' Theorem FAQs
When do I add and when do I subtract in Pythagoras?
Add the squares of the two shorter sides when the hypotenuse is missing. Subtract the square of the known shorter side from the square of the hypotenuse when a shorter side is missing. Checking that your answer is smaller than the hypotenuse confirms you chose correctly.
How do I use Pythagoras in three dimensions?
Apply it twice. First find the diagonal across the base using two of the dimensions, then use that diagonal together with the height as the two shorter sides of a second right-angled triangle. In one step, the space diagonal is the square root of the sum of all three squares.
What are Pythagorean triples?
Sets of three whole numbers that satisfy the theorem, such as 3, 4, 5 and 5, 12, 13 and 8, 15, 17. Any multiple of a triple is also a triple, so 9, 12, 15 works too. Recognising them lets you write down answers without calculating.
Can I use Pythagoras on any triangle?
No. The theorem applies only to right-angled triangles. For a triangle with no right angle you need the cosine rule, or you can split the triangle into two right-angled triangles by dropping a perpendicular from one vertex.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.