Estimating square roots and recognising pi | Year 8

01 / LEARN

The idea, step by step

Trap the root between squares

For nonnegative numbers, squaring preserves order. Since $25 < 30 < 36$ , the positive square root of 30 lies between 5 and 6. To improve the interval, square suitable tenths and compare them with 30. A bound is an interval, not a claim that the root is exactly one of its endpoints.

Pi is a constant ratio

Every circle has circumference divided by diameter equal to $π$ . Thus $C = π d = 2 π r$ . Measured ratios may be close to, rather than exactly, pi because measurement is imperfect. Leave pi in an exact answer; use the specified approximation only when a decimal is requested. A circle with twice the diameter also has twice the circumference.

A common trap

The square root of 30 is halfway between 5 and 6 because 30 is between 25 and 36.

Instead: Squaring is not linear. Use decimal squares or a calculator when a closer estimate is needed.

02 / WORKED EXAMPLES

See the thinking

Example 1

A tenths-wide interval

Find consecutive tenths between which $\sqrt{20}$ lies.

${4.4}^{2} = 19.36$ and ${4.5}^{2} = 20.25$ .
Because $19.36 < 20 < 20.25$ , the root lies between 4.4 and 4.5.

4.4 < \sqrt{20} < 4.5

Example 2

Keep pi exact

A circle has radius 4 cm. Find its exact circumference and its circumference to two decimal places.

Diameter $= 2 \times 4 = 8$ cm, so $C = 8 π$ cm.
Using $π \approx 3.14159$ , $C \approx 25.13272$ cm.
Round only the final value.

8 π

cm, approximately 25.13 cm.

Check your understanding

Between which two consecutive integers is $\sqrt{50}$ ?

Show answer

7 and 8.

$7^{2} = 49 < 50 < 64 = 8^{2}$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Place $\sqrt{35}$ between consecutive integers.
2 marks
Place $\sqrt{70}$ between consecutive integers.
2 marks
Use ${3.4}^{2} = 11.56$ and ${3.5}^{2} = 12.25$ to bound $\sqrt{12}$ .
2 marks
A circle has diameter 6 cm. Give its exact circumference, then use pi = 3.142 to estimate it to two decimal places.
3 marks
Explain why a measured circumference-to-diameter ratio of 3.14 does not prove pi equals 3.14.
2 marks
Which is larger: $\sqrt{26}$ or 5.1? Justify without finding the root.
3 marks

Read the worked solutions

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

Place $\sqrt{35}$ between consecutive integers.
2 marks
$25 < 35 < 36$ .
$5 < \sqrt{35} < 6$
Place $\sqrt{70}$ between consecutive integers.
2 marks
$64 < 70 < 81$ .
$8 < \sqrt{70} < 9$
Use ${3.4}^{2} = 11.56$ and ${3.5}^{2} = 12.25$ to bound $\sqrt{12}$ .
2 marks
Twelve lies between the supplied squares.
$3.4 < \sqrt{12} < 3.5$
A circle has diameter 6 cm. Give its exact circumference, then use pi = 3.142 to estimate it to two decimal places.
3 marks
$C = 6 π$ .
Using the supplied approximation, $6 \times 3.142 = 18.852$ , which rounds to 18.85.
$6 π$ cm; 18.85 cm.
Explain why a measured circumference-to-diameter ratio of 3.14 does not prove pi equals 3.14.
2 marks
Pi is an exact irrational constant.
Finite measurement precision cannot establish its full value.
Measurements and the reported ratio are approximate.
Which is larger: $\sqrt{26}$ or 5.1? Justify without finding the root.
3 marks
${5.1}^{2} = 26.01 > 26$ .
Both values are positive, so $5.1 > \sqrt{26}$ .
5.1

Checking saved progress…

Continue learning

Choosing efficient exact/decimal representations

Next lesson

Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Four operations with integers; signed fraction/decimal calculations; order of operations; terminating and recurring decimals; rational and irrational numbers; estimating square roots and recognising pi; choosing efficient exact/decimal representations.

Official curriculum source ↗

Back to Integers and rational numbers