Once we know how to work out the area of rectangles, there are some handy things we can do. The distributive property of area means we can work out the area of a rectangle by breaking it into smaller rectangles.

To see how we can do this, have a look at Video 1.

Is there only one possible answer?

No! If you imagine the same rectangle that we used in Video 1, see if you can make some different sizes for your smaller rectangles.

		Can you make a different one?
$8\times4$8×4	$3\times4+5\times4$3×4+5×4

Which of these make sense?

Sometimes we might start with two rectangles and want to write a number sentence that matches the picture to calculate the total area. We can do this, but only when certain things are true. Let's see how we can do this in Video 2.

Why break up the rectangles?

Have you wondered why we'd even go to the trouble of breaking our area into two smaller parts to solve? There are a few reasons, but one that you may find useful is in this video. It's short and sharp, and you'll learn a new trick for solving some of your times tables by watching it.

Worked Examples

Question 1

The rectangle below has been split in to two rectangles. We want to work out the area.

$4$4 m

$7$7 m

What is the area of the $purple$ rectangle?
What is the area of the $green$ rectangle?
Write an expression for the area of the whole rectangle:

Area $=$= $\editable{}\times\editable{}$×
What is the area of the whole rectangle?
Is the area of the large rectangle equal to the sum of the two smaller rectangles?

That is, does $7\times4+7\times7=7\times11$7×4+7×7=7×11?
No
A
Yes
B

Question 2

We want to work out the area of the whole rectangle by splitting it into two smaller rectangles.

Which two smaller rectangles can be used to make the larger one?
A
B
C
D
What is the area of the blue rectangle from the answer to part (a)?
What is the area of the orange rectangle from the answer to part (a)?
Write an expression for the area of the whole rectangle:

Area $=$= $\editable{}\times\editable{}$× m²
One of your classmates says:

"I can work out $8\times19$8×19 by adding $8\times9$8×9 and $8\times10$8×10 together."

Are they correct?
Yes
A
No
B

Question 3

We can work out the area of a shape by first splitting it into smaller shapes and then finding the sum of the smaller shape's areas.

Consider a rectangle that measures $7$7 cm by $11$11 cm.

Which of the following are the dimensions of two rectangles that can be combined to make a rectangle measuring $11$11 cm by $7$7 cm?
Rectangle 1 measuring $7$7 cm by $3$3 cm, and

Rectangle 2 measuring $8$8 cm by $7$7 cm.

A
Rectangle 1 with a length of $5$5 and width of $2$2, and

Rectangle 2 with a length of $8$8 and width of $3$3.

B
Rectangle 1 with a length of $7$7 and width of $7$7, and

Rectangle 2 with a length of $11$11 and width of $11$11.

C
Rectangle 1 with a length of $3$3 and width of $2$2, and

Rectangle 2 with a length of $8$8 and width of $5$5.

D
What is the area of Rectangle 1?
What is the area of Rectangle 2?
What is the area of the whole rectangle?
The area of the larger rectangle is equal to the sum of the two smaller rectangles.

Use this to fill in the boxes and complete the number statement: $7\times\editable{}+\editable{}\times8=\editable{}\times11$7×+×8=×11

Outcomes

5.M2.05

Determine, through investigation using a variety of tools (e.g., concrete materials, dynamic geometry software, grid paper) and strategies (e.g., building arrays), the relationships between the length and width of a rectangle and its area and perimeter, and generalize to develop the formulas [i.e., Area = length x width; Perimeter = (2 x length) + (2 x width)]

5.M2.06

Solve problems requiring the estimation and calculation of perimeters and areas of rectangles

Area of rectangles and distributive property