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14.03 Rectangles and squares

Lesson

Introduction

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.

A rectangle with its length and width labelled. Opposite sides are marked as equal, and all corners are right angles.

A rectangle has two pairs of equal opposite sides.

A rectangle with its side labelled. All sides are marked as equal, and all corners are right angles.

A square is a special rectangle with four equal sides.

Area of rectangles and 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 unit, and a single unit square has an area of 1 \text{ unit}^2. In this way, counting the number of unit squares in a shape tells us the area of that shape as a multiple of 1 \text{ unit}^2.

A unit square on a grid with an area of 1 unit squared and a side length of 1 unit.

Let's look at an example.

A rectangle with a length of 4 centimetres and a width of 2 centimetres that is broken up into 8 unit squares.

The rectangle to the left has length 4 cm and width 2 cm and is divided into unit squares, each with a side length of 1 cm. Since the units of the dimensions are in centimetres, the area will be expressed in \text{cm}^2.

By counting, we can see there is a total of 8 unit squares.

So this rectangle has an area of 8 \text{ cm}^2.

Can you see how we can use the dimensions 4 cm and 2 cm to find the area?

Similarly, this method can be used to find the area of a square.

A square that has a side length of 5 metres that is broken up into 25 unit squares.

This square with side length 5 m is divided into unit squares, each with a side length of 1 m. Since the units of the dimensions are in metres, the area will be expressed in \text{m}^2.

By counting, we can see there is a total of 25 unit squares.

So this square has an area of 25\text{ m}^2.

Can you see how we can use the dimensions of the square to find its area?

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 \text{ mm}^2. What rectangles could we create using unit squares that have this area?

We can start with 4 identical unit squares, each with side length 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 \text{ mm}^2.

A rectangle with a length of 1 millimetre and a width of 4 millimetres built from 4 unit squares.

This rectangle has an area of 1 \text{ mm} \times 4 \text{ mm}= 4 \text{ mm}^2.

A rectangle with a length of 4 millimetres and a width of 1 millimetre built from 4 unit squares.

This rectangle has an area of 4 \text{ mm} \times 1 \text{ mm}= 4 \text{ mm}^2.

A rectangle with a length of 2 millimetres and a width of 2 millimetres built from 4 unit squares.

This rectangle has an area of 2 \text{ mm} \times 2 \text{ mm}= 4 \text{ mm}^2.

We can draw three rectangles because there are three pairs of whole numbers that multiply to give an answer of 4 \text{ mm}^2. Notice that 1 \times 4 is the same as 4 \times 1, so the first two rectangles are the same, but one is a rotated version of the other.

Exploration

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 Next Game.

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

Loading interactive...

We can draw different rectangles with different dimensions but with the same area. As we increase the number of unit squares, the area of the rectangle also increases.

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.

The area of a rectangle is given by\text{Area} = \text{length} \times \text{width} \text{, or }\\ \\ A = l \times w

The area of a square is given by\text{Area} = \text{side} \times \text{side} \text{, or }\\ \\ A =s \times s=s^2

Examples

Example 1

Find the area of the rectangle shown.

A rectangle with length of 12 centrimetres and a width of 9 centrimetres.
Worked Solution
Create a strategy

Use the area of a rectangle formula.

Apply the idea
\displaystyle A\displaystyle =\displaystyle l\times wWrite the formula
\displaystyle =\displaystyle 12\times 9Substitute the values
\displaystyle =\displaystyle 108 \text{ cm}^2Evaluate

Example 2

Find the area of the square shown.

A square with side length of 8 centrimetres.
Worked Solution
Create a strategy

Use the formula for the area of a square.

Apply the idea
\displaystyle A\displaystyle =\displaystyle s\times sWrite the formula
\displaystyle =\displaystyle 8\times 8Substitute the values
\displaystyle =\displaystyle 64 \text{ cm}^2Evaluate
Idea summary

The area of the rectangle is given by:

\displaystyle A=l\times w
\bm{A}
is the area of the rectangle
\bm{l}
is the length of the rectangle
\bm{w}
is the width of the rectangle

The area of the square is given by:

\displaystyle A=s\times s
\bm{A}
is the area of the square
\bm{s}
is the side length of the square

Find dimensions from the area

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.

A rectangle with an area of 28 kilometres squared, a length of 7 kilometres, and an unknown width.

The rectangle has an area of 28 \text{ km}^2, and a length of 7 km. How can we determine the width of the rectangle?

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

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

Examples

Example 3

Find the width of this rectangle if its area is 66 \text{ m}^2 and its length is 11 m.

A rectangle with an area of 66 squared metres and a length of 11 metres.
Worked Solution
Create a strategy

Use the formula for the area of a rectangle.

Apply the idea
\displaystyle A\displaystyle =\displaystyle l\times wWrite the formula
\displaystyle 66\displaystyle =\displaystyle 11\times wSubstitute the values
\displaystyle \dfrac{66}{11}\displaystyle =\displaystyle \dfrac{11 \times w}{11}Divide both sides by 11
\displaystyle w\displaystyle =\displaystyle 6 \text{ m}Evaluate
Idea summary

We can find a missing dimension of a rectangle by substituting the area and given dimension into the formula for the area of a rectangle: A=l\times w and then solving for the unknown variable.

Outcomes

MA4-13MG

uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area

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