GCSE Mathematics
Pythagoras' Theorem Practice Questions and Answers
Twelve questions finding the hypotenuse, a shorter side, a ladder height, a rectangle diagonal and a cuboid space diagonal, each naming the hypotenuse first.
The short answer
Twelve questions on Pythagoras' theorem: four finding the hypotenuse, three finding a shorter side, then a ladder, a rectangle diagonal, a coordinate distance, a right-angle test and one three-dimensional space diagonal. Every solution names which side is the hypotenuse before any arithmetic begins.
The questions
- Question 1Foundation2 marks
A right-angled triangle has shorter sides of 3 cm and 4 cm. Find the length of the hypotenuse.
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5 cm
- The unknown side is opposite the right angle, so it is the hypotenuse.
- The hypotenuse is being found, so add: 3² + 4² = 9 + 16 = 25.
- √25 = 5
- The hypotenuse is 5 cm, longer than both given sides, as it has to be.
- Question 2Foundation2 marks
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
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10 cm
- The hypotenuse is the unknown side, opposite the right angle.
- 6² + 8² = 36 + 64 = 100
- √100 = 10, so the hypotenuse is 10 cm.
- This is the 3, 4, 5 triangle with every side doubled.
- Question 3Foundation2 marks
A right-angled triangle has shorter sides of 5 cm and 12 cm. Find the hypotenuse.
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13 cm
- The hypotenuse is unknown, so add the squares of the two shorter sides.
- 5² + 12² = 25 + 144 = 169
- √169 = 13
- The hypotenuse is 13 cm.
- Question 4Foundation2 marks
A right-angled triangle has a hypotenuse of 10 cm and one shorter side of 6 cm. Find the other shorter side.
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8 cm
- The 10 cm side is opposite the right angle, so the hypotenuse is known.
- Because the hypotenuse is known, subtract: 10² - 6² = 100 - 36 = 64.
- √64 = 8, so the missing side is 8 cm.
- Check: 6² + 8² = 36 + 64 = 100 = 10².
- Question 5Core3 marks
A right-angled triangle has shorter sides of 7 cm and 9 cm. Find the hypotenuse, correct to 1 decimal place.
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11.4 cm
- The hypotenuse is unknown, so add.
- 7² + 9² = 49 + 81 = 130
- √130 = 11.4017...
- The hypotenuse is 11.4 cm to 1 d.p. Do not round 130 before square-rooting it.
- Question 6Core3 marks
A right-angled triangle has a hypotenuse of 15 cm and one shorter side of 9 cm. Find the third side.
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12 cm
- The 15 cm side is opposite the right angle and is the longest, so it is the hypotenuse.
- The hypotenuse is known, so subtract: 15² - 9² = 225 - 81 = 144.
- √144 = 12
- The third side is 12 cm - the 3, 4, 5 triangle scaled by 3.
- Question 7Core3 marks
A right-angled triangle has a hypotenuse of 20 cm and one shorter side of 14 cm. Find the third side, correct to 1 decimal place.
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14.3 cm
- 20 cm is the hypotenuse, so subtract the smaller square from the larger.
- 20² - 14² = 400 - 196 = 204
- √204 = 14.2828...
- The third side is 14.3 cm to 1 d.p. It is very close to 14 cm, and still shorter than the 20 cm hypotenuse.
- Question 8Core3 marks
A 5 m ladder leans against a vertical wall. The foot of the ladder is 1.4 m from the base of the wall. How far up the wall does the ladder reach?
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4.8 m
- The wall meets the ground at a right angle, so the ladder is the sloping side: the ladder is the hypotenuse.
- The hypotenuse is known at 5 m, so subtract: 5² - 1.4² = 25 - 1.96 = 23.04.
- √23.04 = 4.8
- The ladder reaches 4.8 m up the wall.
- Question 9Core3 marks
A rectangle measures 12 cm by 9 cm. Find the length of its diagonal.
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15 cm
- The diagonal splits the rectangle into two right-angled triangles with legs 12 cm and 9 cm.
- The diagonal is opposite the right angle at the corner, so it is the hypotenuse.
- 12² + 9² = 144 + 81 = 225
- √225 = 15, so the diagonal is 15 cm.
- Question 10Stretch3 marks
A triangle has sides of 8 cm, 15 cm and 17 cm. Show whether or not it is right-angled.
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Yes - 8² + 15² = 289 = 17², so it is right-angled
- If the triangle is right-angled, the longest side, 17 cm, must be the hypotenuse.
- Add the squares of the other two: 8² + 15² = 64 + 225 = 289.
- Square the longest side: 17² = 289.
- The two agree, so the triangle is right-angled, with the right angle between the 8 cm and 15 cm sides.
- Question 11Stretch3 marks
Find the distance between the points A(1, 2) and B(7, 10).
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10 units
- Draw a horizontal and a vertical line to make a right-angled triangle. AB is the sloping side, so AB is the hypotenuse.
- Horizontal difference: 7 - 1 = 6. Vertical difference: 10 - 2 = 8.
- 6² + 8² = 36 + 64 = 100
- AB = √100 = 10 units.
- Question 12Stretch5 marks
A cuboid measures 6 cm by 8 cm by 24 cm. Find the length of the longest straight rod that will fit inside it.
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26 cm
- The longest rod runs along the space diagonal, corner to opposite corner. Do the base first.
- The base is 6 cm by 8 cm, and its diagonal is the hypotenuse of that triangle: 6² + 8² = 36 + 64 = 100, so the base diagonal is exactly 10 cm.
- The second triangle stands upright: the 10 cm base diagonal and the 24 cm vertical edge are the shorter sides, and the space diagonal is its hypotenuse.
- 10² + 24² = 100 + 576 = 676
- √676 = 26, so the longest rod is 26 cm.
Where these go wrong
- Adding the squares when the hypotenuse was one of the sides given. Adding always yields something longer than both, which cannot be a shorter side.
- Stopping at 130 and writing 130 cm, having squared and added but never square-rooted.
- Subtracting the wrong way round - 196 - 400 rather than 400 - 196 - and then being stuck with a negative.
- Assuming the side drawn sloping is the hypotenuse. It is the side opposite the right angle, whichever way the diagram has been rotated.
- Rounding the base diagonal in the cuboid question and carrying the rounded figure into the second calculation.
Name the hypotenuse before you touch a number
The hypotenuse is the side opposite the right angle, and it is always the longest of the three. That is the whole of the identification, and it does not depend on how the triangle is drawn. A triangle rotated on the page still has its hypotenuse opposite the right angle, even when that side is now horizontal.
Every worked solution below starts by saying which side is the hypotenuse, because that single sentence decides everything that follows. In a word problem the sentence is usually easy: the ladder is the hypotenuse, the diagonal of the rectangle is the hypotenuse, the wire from the top of the mast to the ground is the hypotenuse.
Add or subtract: the only real decision
If the hypotenuse is the side you are looking for, add the squares of the other two. If the hypotenuse is one of the sides you have been given, subtract - the smaller square from the larger, always in that order.
There is a sense check that costs nothing. Adding produces a side longer than both of the others, which is right only if you were finding the hypotenuse. If the answer to a shorter-side question comes out bigger than the hypotenuse you were given, the operation was wrong, and you know it before you have looked at the answers.
A 3-D question is two 2-D questions
The space diagonal of a cuboid looks like a new topic and is not. Find the diagonal across the base first, using the two base edges. That diagonal and the vertical edge then form a second right-angled triangle, standing upright inside the solid, and its hypotenuse is the space diagonal.
The one thing that goes wrong is rounding in between. In the last question the base diagonal is exactly 10 cm, so nothing is lost. When it comes out as 10.63, using 10.6 in the second calculation shifts the final answer, and marks go with it. Keep the squared value - 100, or 113 - and carry that into the second stage instead.
Common questions
How do I identify the hypotenuse?
It is the side opposite the right angle, and it is always the longest side of the three. Find the right angle first and follow the line straight across from it. In word problems the ladder, the diagonal, the wire and the ramp are the hypotenuse, because each of them faces the right angle formed by the ground and the upright.
When do I add and when do I subtract?
Add when you are finding the hypotenuse, subtract when you already have it. One sense check catches every mistake here: the hypotenuse must be longer than the other two sides, so a shorter-side answer that comes out longer than the given hypotenuse means you added when you should have subtracted.
Do I need to memorise the Pythagorean triples?
You do not need to, but recognising 3-4-5, 5-12-13 and 8-15-17 saves time and warns you when an answer is wrong. Their multiples count too: 6-8-10 and 9-12-15 are both the 3-4-5 triangle scaled up, which is why two of the questions above give exact whole-number answers.
Does Pythagoras' theorem work on any triangle?
No. It holds only for right-angled triangles, and applying it to a triangle without a right angle gives a wrong answer that looks perfectly reasonable. For a triangle with no right angle you need the sine rule or the cosine rule instead, which is a separate topic taken later.
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