Finding the hypotenuse | Year 8

01 / LEARN

Understand the idea.

Use addition for the longest side

When both legs are known, use addition of squares to find the hypotenuse square. Then take a square root to obtain its length. Choose the positive root because a triangle side has positive length.

Convert lengths to a common unit before squaring. For example, $400 cm = 4 m$ , so it can be combined with another length in metres. Mixing unconverted centimetres and metres gives the wrong sum.

Check size and units

The hypotenuse must be longer than either leg but shorter than their sum. This provides a useful size check: with legs $3 cm$ and $4 cm$ , a claimed hypotenuse of $25 cm$ is impossible.

Track what the calculation represents. A squared length has square units; its square root returns to ordinary length units. Replacing $c^{2}$ by $c$ without taking that root confuses area with length.

c = \sqrt{a^{2} + b^{2}}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A familiar triple

Find the hypotenuse for legs 9 cm and 12 cm.

Worked solution

1
Identify the unknown as the hypotenuse.
$c^{2} = 9^{2} + 12^{2}$
The two given lengths are legs, so their squares are added.
2
Calculate the squared length.
$c^{2} = 81 + 144 = 225$
Squaring each leg first gives the required square-area sum.
3
Take the positive square root.
$c = \sqrt{225} = 15 cm$
A side length is positive, and the square root returns from square units to centimetres.
4
Check the length's size.
$12 < 15 < 9 + 12$
The result exceeds the longer leg and is below the sum of the legs.

Answer

15 cm.

EXAMPLE 02

Convert first

A right triangle has legs 3 m and 400 cm. Find its hypotenuse.

Worked solution

1
Convert the lengths to metres.
$400 cm = 4 m$
Both legs must use the same unit before their squares are combined.
2
Square and add the leg lengths.
$c^{2} = 3^{2} + 4^{2} = 9 + 16 = 25$
The squared values are now consistently measured in square metres.
3
Take the positive root and state the length.
$c = \sqrt{25} = 5 m = 500 cm$
Taking the root produces the hypotenuse length, which may be stated in either equivalent unit.

Answer

5 m, or 500 cm.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find c for legs 5 cm and 12 cm.

Show answer & reasoning

13 cm.

Square and add the legs: $c^{2} = 5^{2} + 12^{2} = 25 + 144 = 169$ . Taking the positive root gives $c = 13 cm$ , which is longer than either leg.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the hypotenuse for legs 6 cm and 8 cm.
2 marks
Find the hypotenuse for legs 8 m and 15 m.
2 marks
Find the hypotenuse for legs 7 cm and 24 cm.
2 marks
Find the hypotenuse for legs 1.5 m and 2 m.
2 marks
Find the hypotenuse for legs 60 cm and 0.8 m.
3 marks
A student finds $c = 7$ cm for legs 8 cm and 15 cm. Explain the size check and correct the answer.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the hypotenuse for legs 6 cm and 8 cm.
2 marks
$c^{2} = 36 + 64 = 100$ .
Take the positive square root.
10 cm.
Find the hypotenuse for legs 8 m and 15 m.
2 marks
$c^{2} = 64 + 225 = 289$ .
The positive root is 17.
17 m.
Find the hypotenuse for legs 7 cm and 24 cm.
2 marks
$c^{2} = 49 + 576 = 625$ .
The positive root is 25.
25 cm.
Find the hypotenuse for legs 1.5 m and 2 m.
2 marks
$c^{2} = 2.25 + 4 = 6.25$ .
The positive root is 2.5.
2.5 m.
Find the hypotenuse for legs 60 cm and 0.8 m.
3 marks
Convert 60 cm to 0.6 m.
$c^{2} = 0.36 + 0.64 = 1$ .
1 m, or 100 cm.
A student finds $c = 7$ cm for legs 8 cm and 15 cm. Explain the size check and correct the answer.
3 marks
The size check already rejects 7.
Compute $64 + 225 = 289$ and take the root.
The hypotenuse cannot be shorter than either leg; $c = 17$ cm.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Hypotenuse and right-angled triangles; why the theorem works; finding the hypotenuse; finding a shorter side; exact roots versus rounded answers; checking whether a triangle is right-angled; practical length problems.

Official curriculum source ↗

In this lesson

Understand the idea.

Use addition for the longest side

Check size and units

See the method in action.

A familiar triple

Identify the unknown as the hypotenuse.

Calculate the squared length.

Take the positive square root.

Check the length's size.

Convert first

Convert the lengths to metres.

Square and add the leg lengths.

Take the positive root and state the length.

Try it yourself.

Check your understanding

Keep practising