B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Square Roots of Perfect Squares

B1.2About 40 minutes

Learning goals

I can explain that finding a square root undoes squaring.
I can use the √ symbol and find the square root of a perfect square.
I can find the side length of a square when I know its area.
I can use benchmarks, last digits or prime factors to find bigger square roots.

Words to know

square root
A number that, multiplied by itself, gives the number you started with. The square root of 49 is 7, because 7 × 7 = 49.

√ (radical sign)
The symbol for square root. √49 = 7 is read "the square root of 49 is 7".

inverse operations
Operations that undo each other. Squaring and taking the square root undo each other.

benchmark
A friendly number you know well and use to compare, like 20² = 400 and 30² = 900.

Remember

In the last lesson you learned that a perfect square is a whole number times itself, like 49 = 7 × 7 = 7². Today you will work backwards.

A backyard rink

Every winter, Nadia's family floods a square part of their backyard to make a skating rink. This year it covers 64 m². They need boards along each side. How long is each side?

They need a number that, multiplied by itself, makes 64. That number is 8, because 8 × 8 = 64. Each side is 8 m long.

A square backyard ice rink with wooden boards on all four sides, in a snowy yard, with a pair of skates on the boards. — A square rink has four equal sides. If you know its area, the square root tells you how long each side is. AI-generated image

Square roots

The square root of a number is the number that, multiplied by itself, gives the number.

8 × 8 = 64, so the square root of 64 is 8. We write √64 = 8.
Squaring and taking a square root undo each other: 8² = 64 and √64 = 8.

In a square, the area is a perfect square and the side length is its square root.

A square with an area of 64 square units has sides of √64 = 8 units.

Worked example

Example 1: Use a fact you know

Find √81.

Which number times itself is 81?
9 × 9 = 81.

√81 = 9. Check: 9² = 81 ✓

Worked example

Example 2: The rink boards

A square rink has an area of 144 m². How long is each side? How many metres of boards go all the way around it?

Side = √144. 12 × 12 = 144, so each side is 12 m.
There are 4 sides: 4 × 12 = 48 m.

Each side is 12 m, and the boards are 48 m long in total.

Worked example

Example 3: Benchmarks and the last digit

Find √576.

Benchmarks: 20² = 400 and 30² = 900. 576 is between them, so √576 is between 20 and 30.
Last digit: 576 ends in 6. Only numbers ending in 4 or 6 have squares ending in 6 (4 × 4 = 16, 6 × 6 = 36). So try 24 or 26.
Check: 24 × 24 = 480 + 96 = 576 ✓

√576 = 24

Benchmarks trap the square root: 576 sits between 20² = 400 and 30² = 900, so √576 is in the 20s. The last digit narrows it to 24 or 26.

Worked example

Example 4: Prime factors

Find √324 using prime factorization.

Factor tree: 324 = 4 × 81 = 2 × 2 × 3 × 3 × 3 × 3.
Split the factors into two identical groups: (2 × 3 × 3) × (2 × 3 × 3).
Each group is 2 × 3 × 3 = 18.

√324 = 18. Check: 18 × 18 = 180 + 144 = 324 ✓

What about numbers that aren't perfect squares?

Only perfect squares have whole-number square roots. √50 is not a whole number: it is between √49 = 7 and √64 = 8. You will learn to estimate square roots like this in Grade 8. For now, if a square root isn't a whole number, say which two whole numbers it is between.

Hands-on activity

Square root detective

You need: a calculator (a phone calculator is fine), paper and a pencil. Play with a grown-up.

Your grown-up secretly picks a whole number from 11 to 30, squares it on the calculator, and tells you only the answer (for example, "529").
You find the square root without using a √ key. Use benchmarks (10² = 100, 20² = 400, 30² = 900) and the last-digit trick, then check your guess by multiplying.
Count how many guesses you needed. Then switch roles.

Example: 529 is between 400 and 900, so the root is in the 20s. It ends in 9, so try 23 or 27. 23 × 23 = 529 ✓

Bonus: Draw a square with an area of 16 cm² and one with an area of 25 cm² on paper using a ruler. What side lengths did you use? (4 cm and 5 cm.)

Just for fun

Why are square roots so good at solving mysteries?

They always get to the root of the problem.

Your turn

Find √25.

Find √100.

Find √169.

A square tablecloth has an area of 4 m². How long is each side?

Find √256.

A square garden has an area of 121 m². How long is each side? What is the perimeter?

Is √40 a whole number? If not, which two whole numbers is it between?

More practice: warm-up

Find √9.

Find √36.

Find √64.

Find √1.

Find √144.

More practice: on your own

Find √400.

Find √225.

A square quilt is made of 196 small square patches. How many patches are along each side?

Find √441.

Which is larger: √81 or 10?

Challenge

Find √784 using benchmarks and the last digit.

576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3. Use the prime factors to find √576.

A square photo has an area of 900 cm². A frame adds 5 cm on every side. What is the area of the photo and frame together?

Check your understanding

Quiz

What is √49?

A square has an area of 64 cm². How long is each side?

Which statement is true?

What is √196?

Which number has a whole-number square root?

30² = 900. So what is √900?

For parents and teachers

Big idea: This lesson finishes B1.2: determine the square roots of perfect squares in various contexts (area of squares, square arrays). Estimating square roots of numbers that are not perfect squares is a Grade 8 expectation, so we only mention it.

Common difficulties:

Dividing by 2 instead of taking the square root (√64 = 32). Ask: "What number times itself makes 64?"
Mixing up area and side length. Draw the square and label both.
Giving up on bigger numbers. Teach the three-step plan: benchmark (which tens?), last digit (which ones?), check by multiplying.

How to help: Play square root detective. A calculator is a great checking tool here, but encourage your child to reason first and use it only to check.