Year 8 - Pythagoras' theorem - Free practice

Year 8 Pythagoras' Theorem Worksheets

Find the hypotenuse, find a shorter side, apply to word problems.

Quick answer

How do you use Pythagoras' theorem to find a missing side of a right-angled triangle?

Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse. To find a missing side, substitute the known sides and solve. For example, if $a = 3$ and $b = 4$ , then $c = \sqrt{9 + 16} = 5$ .

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Worked examples

Year 8 pythagoras' theorem: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Finding the hypotenuse
A right-angled triangle has shorter sides of length $9$ m and $12$ m. Find the length of the hypotenuse.
Show worked solution
Key idea: In a right-angled triangle, $c^2 = a^2 + b^2$ , where $c$ is the hypotenuse.
1. Substitute the known sides: $c^2 = 9^2 + 12^2$
2. Evaluate: $c^2 = 81 + 144 = 225$
3. Take the square root: $c = \sqrt{225} = 15$
Answer: The hypotenuse is $15$ m.
Example 2 · Finding a shorter side
A right-angled triangle has a hypotenuse of $10$ m and one shorter side of $6$ m. Find the exact length of the other shorter side, then round your answer to two decimal places.
Show worked solution
Key idea: Rearrange $a^2 + b^2 = c^2$ to find a shorter side: $b^2 = c^2 - a^2$ .
1. Substitute the known values: $b^2 = 10^2 - 6^2$
2. Evaluate: $b^2 = 100 - 36 = 64$
3. Take the square root: $b = \sqrt{64} = 8$
4. The exact answer is $8$ m, which rounds to $8.00$ m.
Answer: The other shorter side is exactly $8$ m (or $8.00$ m to two decimal places).
Example 3 · Ladder reaching a wall
A ladder is leaning against a vertical wall. The base of the ladder is $1.5$ m from the wall along flat ground, and the ladder reaches $3.6$ m up the wall. Find the length of the ladder, correct to two decimal places.
Show worked solution
Key idea: The ladder, wall, and ground form a right-angled triangle. The ladder is the hypotenuse.
1. Substitute into $c^2 = a^2 + b^2$ : $c^2 = 1.5^2 + 3.6^2$
2. Evaluate: $c^2 = 2.25 + 12.96 = 15.21$
3. Take the square root: $c = \sqrt{15.21} \approx 3.90$
Answer: The ladder is approximately $3.90$ m long.

Common mistakes

Where students usually lose marks

Adding instead of subtracting when finding a shorter side

Student writes $b^2 = 10^2 + 6^2 = 136$ , even though the hypotenuse is known.

When finding a shorter side, subtract the square of the known shorter side from the square of the hypotenuse: $b^2 = c^2 - a^2$ . Only add when finding the hypotenuse.

Choosing the wrong side as the hypotenuse

Student uses the longest given side as $a$ or $b$ rather than as $c$ .

The hypotenuse is always the side opposite the right angle, and it is always the longest side. Label $c$ first, then assign $a$ and $b$ to the other two sides.

Forgetting to take the square root

Student writes $c^2 = 225$ and gives the answer as $225$ m.

After finding $c^2$ , always take the positive square root to get the length: $c = \sqrt{225} = 15$ .

Rounding √(b²) when b is a whole number

Student writes $\sqrt{64} \approx 8.00$ and concludes the answer is not exact.

Check whether the square root is a whole number before rounding. If $b^2 = 64$ , then $b = 8$ exactly, and no rounding is needed (or the rounded form is simply $8.00$ ).

Try one yourself

Your turn

A ladder leans against a wall. The base of the ladder is $1.8$ m from the wall and the ladder reaches $2.4$ m up the wall. How long is the ladder, correct to 2 decimal places? Give your answer in metres.

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About this worksheet

Why we built it

Pythagoras' theorem is one of the most-tested Year 8 topics. These worksheets give your student practice across the three standard question types - finding the hypotenuse, finding a shorter side, and applying Pythagoras to real-world problems like ladders, ramps and TV screen sizes.

What's covered

Sub-skills your student will practise

✓Finding the hypotenuse given two shorter sides
✓Finding a shorter side given the hypotenuse
✓Identifying when a triangle is right-angled from its side lengths
✓Applied problems (ladders, ramps, diagonal distances)
✓Exact (surd) vs decimal answers

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Frequently asked

Questions parents ask about pythagoras' theorem

What is Pythagoras' theorem in plain English?

In a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides. Written as a^2 + b^2 = c^2 where c is the hypotenuse.

How do students find a shorter side, not the hypotenuse?

Rearrange the formula: if c^2 = a^2 + b^2, then a^2 = c^2 - b^2. The shorter side equals the square root of the difference of squares, not the sum.

When does Pythagoras work?

Only in right-angled triangles. If the triangle has no 90-degree angle, Pythagoras doesn't apply - you'd need trigonometry or the cosine rule (Year 9-10 topics).

What real-world problems use Pythagoras?

Ladder problems (how tall a wall can a ladder reach), TV screen sizes (diagonal measurement), ramp gradients, and the distance between two points on a map. Year 8 introduces these contexts so the abstract formula has somewhere to land.

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