Year 7 - Squares and square roots - Free practice

Year 7 Squares and Square Roots Worksheets & Practice

Build fluency with perfect squares and square roots. Worked solutions every time, free.

Quick answer

What are square numbers and square roots?

A perfect square is the result of multiplying a whole number by itself, e.g. 5² = 5 × 5 = 25. The square root reverses this: √25 = 5. To find the square root of a perfect square, ask which number multiplied by itself gives that value.

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Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

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Worked examples

Year 7 squares and square roots: 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 · Squaring a number
Find the value of $8^2$ .
Show worked solution
Key idea: $8^2$ means $8 \times 8$ , not $8 \times 2$ .
1. Write the meaning of the notation: $8^2 = 8 \times 8$
2. Multiply: $8 \times 8 = 64$
Answer: $8^2 = 64$
Example 2 · Finding the square root of a perfect square
Find the value of $\sqrt{144}$ .
Show worked solution
Key idea: $\sqrt{144}$ asks "what number multiplied by itself equals $144$ ?"
1. Recall or check perfect squares: $10^2 = 100$ , $11^2 = 121$ , $12^2 = 144$ .
2. Since $12 \times 12 = 144$ , we have $\sqrt{144} = 12$ .
3. Check by squaring: $12^2 = 144$ ✓
Answer: $\sqrt{144} = 12$
Example 3 · Solving a real-life problem using square roots
A square courtyard has an area of $196$ square metres. What is the side length of the courtyard?
Show worked solution
Key idea: The area of a square is side length squared, so the side length is the square root of the area.
1. Write the relationship: $\text{side length} = \sqrt{\text{area}} = \sqrt{196}$
2. Find which number multiplied by itself gives $196$ : $14 \times 14 = 196$ , so $\sqrt{196} = 14$ .
3. Check: $14^2 = 196$ ✓
Answer: The side length of the courtyard is $14$ metres.

Common mistakes

Where students usually lose marks

Multiplying by 2 instead of by itself

$6^2 = 6 \times 2 = 12$

$6^2$ means $6$ multiplied by itself, not by $2$ . The correct working is $6^2 = 6 \times 6 = 36$ . Read the notation as "six squared", which always means "six groups of six".

Forgetting perfect squares when finding square roots

$\sqrt{81} = 40.5$ (halving $81$ instead of finding the square root)

Finding a square root is not the same as dividing by $2$ . Memorise the perfect squares up to $12^2 = 144$ so you can recognise that $9 \times 9 = 81$ , giving $\sqrt{81} = 9$ .

Thinking every whole number has a whole-number square root

$\sqrt{20} = 10$ (halving $20$ ) or $\sqrt{20} = 4$ (guessing)

Only perfect square numbers have whole-number square roots. Because $20$ is not a perfect square, $\sqrt{20}$ is not a whole number. Check by listing perfect squares: $16, 25, 36, \ldots$ and notice $20$ is not among them.

Confusing the inverse relationship direction

"Squaring and square roots are inverses, so $\sqrt{7^2} = 49$ "

Applying squaring then square rooting (or vice versa) returns you to the starting number, not a new one. $\sqrt{7^2} = \sqrt{49} = 7$ , right back to where you started. The operations undo each other.

Try one yourself

Your turn

A square patio has an area of $144 \text{ m}^2$ . What is the side length of the patio, in metres?

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About this worksheet

Why we built it

Squares and square roots are one of the first topics in Year 7 where students see how multiplication and a new operation undo each other. These worksheets guide students from recognising perfect squares all the way to using square roots to solve simple problems, with worked solutions for every question so they can check their reasoning step by step.

What's covered

Sub-skills your student will practise

✓Identifying perfect square numbers up to at least 12²
✓Understanding square notation, e.g. 5² = 5 × 5 = 25
✓Finding the square root of perfect squares, e.g. √49 = 7
✓Describing the inverse relationship between squaring and square roots
✓Using squares and square roots to solve word problems
✓Ordering and comparing perfect squares on a number line

From our families

Parents on the practice questions.

“Tuterly has been exactly what we needed since Tom started secondary school. Once the topic tests began, he was struggling to keep up. Now he has unlimited practice, all matched to what his class is actually covering, and his Maths results have improved noticeably.”

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Year 8 · Maths

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Year 10 · Maths

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Frequently asked

Questions parents ask about squares and square roots

What is a perfect square and which ones should my Year 7 student know?

A perfect square is the result of multiplying a whole number by itself, for example 36 = 6 × 6. Year 7 students are expected to be fluent with perfect squares from 1² up to at least 12², so from 1 all the way to 144.

How do squares and square roots relate to each other?

Squaring a number and taking its square root are inverse operations, meaning they undo each other. Because 8² = 64, we know √64 = 8. Seeing this two-way relationship is the core idea of this topic in the curriculum.

What kinds of problems will my child solve using squares and square roots?

Students apply squares and square roots to problems such as finding the side length of a square given its area, or working out missing values in simple equations like x² = 81. These problems build the thinking skills needed for geometry and algebra later in Year 7 and beyond.

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