B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Square Roots of Decimals and Fractions, and Square Roots in Real Life

B1.3About 40 minutes

Learning goals

I can find square roots of decimals and fractions that are perfect squares, like √0.49 and √(9/16).
I can estimate square roots of decimals that are not perfect squares.
I can use square roots to find the side of a square or the radius of a circle from its area.
I can use a square-root formula in a real situation.

Words to know

square root of a fraction
If a fraction's numerator and denominator are both perfect squares, take the square root of each: √(9/16) = 3/4.

radius
The distance from the centre of a circle to its edge.

area of a circle
A = π × r × r, or A = πr². To find r from A, divide by π and then take the square root: r = √(A ÷ π).

formula
A rule written with symbols that tells how to calculate something.

Snow day quilt

Grandma is sewing a square quilt that will cover 2.25 m². How long is each side?

We need √2.25. Since 1.5 × 1.5 = 2.25, each side is 1.5 m. Square roots work for decimals and fractions too!

Square roots of decimals

A decimal is a perfect square if it is the square of a decimal. Use what you know about whole numbers:

7 × 7 = 49, so 0.7 × 0.7 = 0.49 and √0.49 = 0.7.
12 × 12 = 144, so 1.2 × 1.2 = 1.44 and √1.44 = 1.2.
3 × 3 = 9, so 0.3 × 0.3 = 0.09 and √0.09 = 0.3 (not 0.03!).

Count the decimal places. Squaring a number with 1 decimal place gives 2 decimal places. Squaring a number with 2 decimal places gives 4. So a perfect-square decimal has an even number of decimal places, and its square root has half as many.

Not every decimal is a perfect square. √0.5 is irrational: 0.7² = 0.49 and 0.71² = 0.504 1, so √0.5 ≈ 0.71.

A surprise: for numbers between 0 and 1, the square root is bigger than the number. √0.25 = 0.5, and 0.5 > 0.25.

The big square is 1 whole, split into 100 hundredths. A 0.3 by 0.3 square covers 9 of them, so 0.3 × 0.3 = 0.09 and √0.09 = 0.3.

Square roots of fractions

If the numerator and the denominator are both perfect squares, take the square root of each:

√(9/16) = 3/4, because 3/4 × 3/4 = 9/16.
√(4/25) = 2/5.

Square roots in real life

Side of a square from its area: side = √area.
Radius of a circle from its area: in Grade 7 you learned that A = πr², so r² = A ÷ π, and r = √(A ÷ π).
Science formulas: for example, if you drop a ball (and ignore air resistance), the time in seconds to fall h metres is about t = √(h ÷ 4.9).

Left: a square with an area of 2.25 m² has sides of √2.25 = 1.5 m. Right: a circle with an area of 50 m² has a radius of about √(50 ÷ π) ≈ 4 m. (Not to scale.)

Worked example

Example 1: Decimals

Find √0.49 and √0.0016.

7 × 7 = 49 and 0.49 has 2 decimal places, so the root has 1: √0.49 = 0.7. Check: 0.7 × 0.7 = 0.49. ✓
4 × 4 = 16 and 0.001 6 has 4 decimal places, so the root has 2: √0.001 6 = 0.04. Check: 0.04 × 0.04 = 0.001 6. ✓

√0.49 = 0.7 and √0.001 6 = 0.04

Worked example

Example 2: A fraction

Find √(25/36).

25 = 5² and 36 = 6².
So √(25/36) = 5/6. Check: 5/6 × 5/6 = 25/36. ✓

√(25/36) = 5/6

Worked example

Example 3: Side and perimeter from area

A square quilt has an area of 2.25 m². How much ribbon is needed to go all the way around its edge?

Side = √2.25 = 1.5 m (15 × 15 = 225).
Perimeter = 4 × 1.5 = 6 m.

6 m of ribbon

Worked example

Example 4: Radius from area

A circular garden has an area of 50 m². About how long is its radius?

Use r = √(A ÷ π).
50 ÷ π ≈ 50 ÷ 3.14 ≈ 15.9.
√15.9 is just under 4, because 4² = 16. A calculator gives about 3.99.

The radius is about 4 m.

Watch out!

√0.09 = 0.3, not 0.03. Check: 0.03 × 0.03 = 0.000 9.
In r = √(A ÷ π), divide first, then take the square root.
√ means the positive square root. (−5) × (−5) = 25 too, but √25 = 5.

Just for fun

Why does √0.09 always feel misunderstood?

Everyone keeps calling it 0.03, but it's really 0.3. (0.03 × 0.03 = 0.000 9, so check before you guess!)

Hands-on activity

Squares and circles in chalk

You need: sidewalk chalk (or painter's tape indoors), a measuring tape, a piece of string about 1 m long, a calculator. Work outdoors on a driveway or sidewalk only with a grown-up's permission, away from traffic.

Draw a square with an area of 0.25 m². What is its side length? Draw a 1 m² square next to it. How many of the small squares would fit in the big one?
Plan a circle with an area of 0.5 m². Find the radius: r = √(0.5 ÷ π).
Hold one end of the string at the centre and tie or hold the chalk at the radius length. Walk around to draw the circle.

What to look for: the 0.25 m² square has sides of √0.25 = 0.5 m, and 4 of them fit in the 1 m² square. For the circle, 0.5 ÷ π ≈ 0.159, and √0.159 ≈ 0.40 m (40 cm).

Math talk: "The small square's area is 1/4 of the big one. Why is its side 1/2, not 1/4?"

Your turn

Find √0.36.

Find √1.21.

Find √(16/49).

Find √0.04.

A square tablecloth has an area of 1.69 m². How long is each side?

Is √0.4 rational? Estimate it to two decimal places.

A square vegetable plot has an area of 18 m². Find each side to the nearest tenth of a metre.

More practice: warm-up

Find √0.81.

Find √0.01.

Find √(1/4).

Find √(9/100).

Find √144.

More practice: on your own

Find √2.56.

Find √0.004 9.

Find √(49/64).

A square patio has an area of 12.25 m². What is its perimeter?

A round pizza has an area of about 700 cm². About how long is its radius?

Challenge

Ignoring air resistance, a dropped ball takes about t = √(h ÷ 4.9) seconds to fall h metres. How long does it take to fall 44.1 m?

One square has 9 times the area of another. How do their side lengths compare? Explain.

Two square rugs have areas of 6.25 m² and 4 m². They are placed side by side, edge to edge, in a straight row. How long is the row?

Check your understanding

Quiz

What is √0.64?

What is √0.09?

What is √(25/81)?

A square has an area of 3.24 m². How long is each side?

A circle has an area of 314 cm². Using π ≈ 3.14, what is its radius?

Which number has an irrational square root?

For parents and teachers

Big idea: This lesson finishes B1.3 (estimate and calculate square roots, in various contexts) by extending square roots to decimals and fractions and using them in real situations: the side of a square, the radius of a circle, and a science formula.

Common difficulties:

Decimal-place errors: √0.09 = 0.3, not 0.03. Have your child square the answer to check.
Being surprised that √0.25 = 0.5 is bigger than 0.25. Talk about why: multiplying by a number less than 1 makes things smaller, so squaring 0.5 shrinks it.
In r = √(A ÷ π), taking the square root before dividing.

How to help: Measure square and round things at home (tiles, a round table) and work backwards from area. If you do the chalk activity, stay with your child, away from traffic.