Grade by Grade
E. Spatial Sense · Angles, Rectangles, Grids and Measurement

Using the Area Formula to Find a Missing Side

E2.6 A1About 35 minutes
Learning goals
  • I can write the area formula for a rectangle: A = l × w.
  • I can find the area when I know the length and the width.
  • I can find a missing length or width when I know the area and the other side.
  • I can check my answer by multiplying.

Words to know

formula
A rule written with letters and symbols, like A = l × w.
A, l and w
Short for area, length and width.
unknown
The number we are trying to find. We can show it with a ? or a box.
missing factor
The number missing in a multiplication, like the ? in 6 × ? = 24. Division helps find it.

Remember from the last lesson

The area of a rectangle is the number of rows times the number of squares in each row, which is the length × width. Mathematicians write this as a short rule called a formula:

A = l × w

(A stands for area, l for length and w for width.)

AreaAlength (l)width (w)A = l × w
The area formula for a rectangle: Area = length × width.

Let's start with a garden

Grandma is planning a vegetable garden. She has enough soil to cover 24 m². She wants the garden to be 6 m long. How wide should it be?

This time we know the area and one side. The width is the unknown. The formula still helps!

Area = 24 m²6 m? m
A garden with an area of 24 m² and a length of 6 m. What is the width?
Worked example

Example 1: Find the missing width

A = 24 m² and l = 6 m. Find w.

  1. Write the formula: A = l × w.
  2. Put in what you know: 24 = 6 × ?
  3. Think: 6 times what makes 24? Or divide: 24 ÷ 6 = 4.
  4. Check: 6 × 4 = 24. ✔

The garden should be 4 m wide.

Why division works

Think of the rows. If a rectangle has an area of 28 squares and there are 4 rows, the 28 squares are shared equally among 4 rows. 28 ÷ 4 = 7 squares in each row. So the other side is 7.

one row of 7 is shown4 rows, area 284 × ? = 2828 ÷ 4 = 7
28 squares in 4 equal rows means 7 squares in each row. So 4 × 7 = 28.
Worked example

Example 2: Find the area

A tile is 8 cm long and 5 cm wide. Find its area.

  1. A = l × w
  2. A = 8 × 5 = 40.

A = 40 cm².

Worked example

Example 3: Find the missing length

A rectangular rug has an area of 18 m². It is 3 m wide. How long is it?

  1. A = l × w, so 18 = ? × 3.
  2. 18 ÷ 3 = 6.
  3. Check: 6 × 3 = 18. ✔

The rug is 6 m long.

Worked example

Example 4: A square

A square sandbox has an area of 49 m². How long is each side?

  1. In a square, the length and width are the same number.
  2. Which number times itself is 49? Try: 6 × 6 = 36 (too small), 7 × 7 = 49. ✔

Each side is 7 m.

Use what you know about fact families

Multiplication and division are partners. If 6 × 4 = 24, then 24 ÷ 6 = 4 and 24 ÷ 4 = 6. So when you know the area and one side, divide the area by that side to find the other side.

Math talk: does it make sense?

Always check by multiplying your answer by the side you were given. It should equal the area. Also check the units: sides are in cm or m, and area is in cm² or m².

Just for fun

Riddle: I am a rectangle with an area of 24 cm². One of my sides is 6 cm. How long is my other side?

Answer: 4 cm, because 6 × 4 = 24. Rectangle mystery solved, Detective!

Hands-on activity

Secret rectangle game

You need: centimetre grid paper (or draw your own grid), a pencil and a ruler. Play with a grown-up.

  1. Player 1 secretly draws a rectangle on grid paper and counts its area.
  2. Player 1 tells the area and one side: "My rectangle has an area of 32 square units. One side is 4."
  3. Player 2 uses the formula to find the other side (32 ÷ 4 = 8), then draws the rectangle.
  4. Compare drawings. Do they match? (Turned rectangles count as a match!)
  5. Switch roles. Play 5 rounds.
  6. Bonus round: give only the area, like 24. How many different rectangles with whole-number sides can you find? (1 by 24, 2 by 12, 3 by 8, 4 by 6.)

Math talk: "How did you know which number to divide by?"

Your turn

1

Find the area: l = 7 cm, w = 3 cm.

2

A rectangle has an area of 20 cm² and a length of 5 cm. What is the width?

3

A rectangle has an area of 36 m² and a width of 4 m. What is the length?

4

A square has an area of 25 cm². How long is each side?

5

A card has an area of 54 cm² and a length of 9 cm. What is its width?

6

Fill in the missing number: A = 40 m², l = ?, w = 8 m.

More practice: warm-up

1

Write the formula for the area of a rectangle.

2

6 × ? = 30. What is the missing number?

3

Find the area: l = 4 m, w = 2 m.

4

A = 12 cm², l = 6 cm. What is w?

5

A = 10 m², w = 1 m. What is l?

More practice: on your own

1

A classroom rug covers 24 m² and is 4 m wide. How long is it?

2

A photo has an area of 63 cm². One side is 7 cm. How long is the other side?

3

A square garden has an area of 64 m². What is the length of each side? What is its perimeter?

4

Sam says a rectangle with area 30 cm² and length 6 cm has a width of 24 cm. What mistake did he make?

5

Which rectangle has the greater area: 5 m by 9 m, or 6 m by 7 m?

Challenge

1

A backyard rink must cover 72 m². It is 6 m wide. How long must it be?

2

List all the rectangles with whole-number sides that have an area of 12 cm².

3

A rectangle has an area of 32 m² and a length of 8 m. Find its width, then its perimeter.

Check your understanding

Quiz

1.

What is the formula for the area of a rectangle?

2.

A rectangle has an area of 35 cm² and a length of 7 cm. What is the width?

3.

A square has an area of 81 m². How long is each side?

4.

To find a missing side when you know the area and one side, you...

5.

A = 48 m² and w = 6 m. What is l?

6.

A field is 10 m long and 9 m wide. What is its area?

For parents and teachers

Big idea: The Ontario curriculum (E2.6) asks Grade 4 students to apply the formula for the area of a rectangle (A = l × w) to find the unknown measurement when given two of the three. When the area and one side are known, the missing side is found by division (or by thinking of a missing factor: 6 × ? = 24). This connects area to multiplication and division facts.

Common difficulties:

  • Subtracting instead of dividing (area 30, side 6, answer 24). Ask: "6 times what makes 30?"
  • Not checking. Encourage multiplying the answer by the known side to see if it gives the area.
  • Unit mix-ups. Sides are in cm or m; area is in cm² or m².

How to help: Practise multiplication and division facts, since they make this lesson much easier. Plan pretend projects together: "We have tiles to cover 20 m². If the patio is 5 m long, how wide can it be?"