Exact roots versus rounded answers | Year 8

01 / LEARN

The idea, step by step

Not every length is a whole number

For legs 1 cm and 2 cm, the hypotenuse is exactly $\sqrt{5}$ cm. A decimal approximation is useful for a measurement, but it is not the same exact value.

Leave a root exact when requested. If a decimal is requested, use the required number of decimal places and include an approximation sign or the word approximately.

Round once at the end

Keep the squared sum or difference exact until you take its root. Early rounding can alter the last digit of a final measurement. Use square-number bounds as an independent check.

For example, a length $\sqrt{50}$ lies between 7 and 8 because 49 < 50 < 64. A scientific calculator can refine that value, but bounds explain its size.

A common trap

Writing a rounded decimal after an equals sign as though it is exact.

Instead: Use an approximation symbol or say approximately; retain the root when exactness is required.

02 / WORKED EXAMPLES

See the thinking

Example 1

Exact and measured forms

Find the hypotenuse of a right triangle with legs 1 cm and 2 cm. Give exact form and 2 decimal places.

$c^{2} = 1 + 4 = 5$ .
Take the positive root.
A calculator gives about 2.236…, which rounds to 2.24.

\sqrt{5}

cm; approximately 2.24 cm.

Example 2

Check a decimal with bounds

A missing leg satisfies $x^{2} = 50$ . Explain its integer bounds and round to 1 decimal place.

Since $7^{2} = 49$ and $8^{2} = 64$ , the root is between them.
The root is about 7.071….
To one decimal place, the second decimal digit rounds the result up.

7 < x < 8; x is approximately 7.1.

Check your understanding

A hypotenuse has $c^{2} = 20$ . Between which consecutive integers is c?

Show answer

4 and 5.

The neighbouring squares are 16 and 25.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Legs are 2 cm and 3 cm. Give the exact hypotenuse.
2 marks
Hypotenuse 6 cm, leg 4 cm: give the other leg exactly.
2 marks
Between which integers is $\sqrt{41}$ ?
2 marks
A calculator gives 9.433981 for a length in metres. Round to 2 decimal places.
2 marks
A calculator gives 6.0827625 for a length in centimetres. Round to 1 decimal place.
2 marks
For legs 5 cm and 7 cm, a calculator gives $\sqrt{74} = 8.602325 \dots$ . Give the exact answer, the answer to 2 decimal places, and an integer-bound check.
3 marks

Read the worked solutions

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

Legs are 2 cm and 3 cm. Give the exact hypotenuse.
2 marks
Add squares: 4 + 9 = 13.
Take the positive root without rounding.
$\sqrt{13}$ cm.
Hypotenuse 6 cm, leg 4 cm: give the other leg exactly.
2 marks
Subtract squares: 36 - 16 = 20.
Take the positive root.
$\sqrt{20}$ cm.
Between which integers is $\sqrt{41}$ ?
2 marks
The neighbouring squares are 36 and 49.
41 lies between them.
6 and 7.
A calculator gives 9.433981 for a length in metres. Round to 2 decimal places.
2 marks
The third decimal digit is 3.
Keep the hundredths digit unchanged.
9.43 m.
A calculator gives 6.0827625 for a length in centimetres. Round to 1 decimal place.
2 marks
The second decimal digit is 8.
Round the tenths digit up.
6.1 cm.
For legs 5 cm and 7 cm, a calculator gives $\sqrt{74} = 8.602325 \dots$ . Give the exact answer, the answer to 2 decimal places, and an integer-bound check.
3 marks
The squared sum is 74.
Use 64 < 74 < 81 for the bounds, then round the supplied value.
$\sqrt{74}$ cm; approximately 8.60 cm; between 8 and 9.

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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 ↗

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