10.04 Area of rectangles and squares

A rectangle has two pairs of equal opposite sides. A square is a special rectangle with four equal sides. Below, a rectangle and a square are drawn with markings on their edges. Edges with the same markings show that they are the same size.

Finding the area by counting unit squares

The area of a rectangle is the amount of space that can fit within its outline. One method of finding the area is to divide a rectangle into unit squares and to count the number of these squares.

A unit square is defined to be a square with a side length of $1$1 unit, and a single unit square has an area of $1$1 unit². In this way, counting the number of unit squares in a shape tells us the area of that shape as a multiple of $1$1 unit².

Let's look at an example. The rectangle below has length $4$4 cm and width $2$2 cm and is divided into unit squares, each with a side length of $1$1 cm. Since the units of the dimensions are in cm, the area will be expressed in cm².

					By counting, we can see there is a total of $8$8 unit squares. So this rectangle has an area of $8$8 cm2. Can you see how we can use the dimensions $4$4 cm and $2$2 cm to find the area?

 

Similarly, this method can be used to find the area of a square. Below, a square with side length $5$5 m is divided into unit squares, each with a side length of $1$1 m. Since the units of the dimensions are in m, the area will be expressed in m².

					By counting, we can see there is a total of $25$25 unit squares. So this square has an area of $25$25 m2. Can you see how we can use the dimensions of the square to find its area?

 

Rectangles with the same area

From the previous section, you may have noticed that multiplying the dimensions of a rectangle gives its area. Are there different pairs of numbers that multiply to give the same answer? In other words, is it possible for rectangles to have the same area when they have different dimensions?

For example, suppose we knew only that a rectangle had an area of $4$4 mm². What rectangles could we create using unit squares that have this area?

We can start with $4$4 identical unit squares, each with side length $1$1 mm, and find the ways we can arrange them so that they form a rectangle. Here are the three rectangles we can form that have whole number side lengths and an area of $4$4 mm².

					This rectangle has an area of $1$1 mm$\times4$×4 mm$=4$=4 mm2.
					This rectangle has an area of $4$4 mm$\times1$×1 mm $=4$=4 mm2.
					This square has an area of $2$2 mm$\times2$×2 mm $=4$=4 mm2.

We can draw three rectangles because there are three pairs of whole numbers that multiply to give an answer of $4$4 mm². Notice that $1\times4$1×4 is the same as $4\times1$4×1, so the first two rectangles are the same, but one is a rotated version of the other.

 

Experimenting with different dimensions that give the same area

Use the applet below to draw rectangles with a target area. The left side of the applet will tell you how many rectangles can be drawn. Once you have drawn the required number of rectangles, click $\editable{\text{Next game}}$Next game.

As an added challenge, you can click $\editable{\text{Hide Grid}}$Hide Grid to draw rectangles by thinking of two numbers that multiply to give the target area, rather than counting the unit squares.

Using dimensions to find the area

While dividing rectangles into unit squares is effective for finding the area, it can be time consuming, especially for larger rectangles. The method of multiplying the dimensions of a rectangle is the quickest way to find its area. The formula for the area of a square is very similar.

 

Worked examples

Example 1

Find the area of the rectangle below.

Think: We know the length and width of the rectangle, so we can use the formula to find the area. The dimensions are given in mm, so the area will be in mm².

Do:

$\text{Area }$Area	$=$=	$\text{length }\times\text{width }$length ×width	(Formula for the area of a rectangle)
	$=$=	$5\times3$5×3	(Substitute the values for the length and width)
	$=$=	$15$15	(Perform the multiplication to find the area)

So this rectangle has an area of $15$15 mm².

Reflect: The length of a rectangle is often defined to be the longest side, which makes the width the shortest side. But since the area formula uses multiplication, the order does not matter. So even if we use another convention that defines the length to be always the horizontal side, the same formula will give the same area.

 

Example 2

Find the area of the square below.

Think: We know the side length of the square, so we can use the formula to find the area. The dimension is given in cm, so the area will be in cm².

Do:

$\text{Area }$Area	$=$=	$\text{side }\times\text{side }$side ×side	(Formula for the area of a square)
	$=$=	$6\times6$6×6	(Substitute the value for the side)
	$=$=	$36$36	(Perform the multiplication to find the area)

So this square has an area of $36$36 cm².

Reflect: The area of a square involves the product of the side length with itself. Another way to write this is using index notation, $\text{Area }=\text{side }^2$Area =side 2. In this example the area would be $6^2=36$62=36 cm².

 

Using the area to find a dimension

We have found that the area of a rectangle is given by the product of its length and width. If we already know the area, along with one of the dimensions, we can use this relationship to find the remaining dimension.

The rectangle below has an area of $28$28 km², and a length of $7$7 km. How can we determine the width of the rectangle?

From the formula, we know that $\text{Area }=\text{length }\times\text{width }$Area =length ×width , which means that $28=7\times\text{width }$28=7×width . So we want to find the number that multiplies with $7$7 to give $28$28.

In other words, we can find the number of times $7$7 fits into $28$28. This is given by $\frac{28}{7}=4$287​=4, so the width of the rectangle is $4$4 km.

 

Practice questions

Question 1

Question 2

Question 3