Topics
10. 2D Measurement
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AustraliaNSW
Stage 3

10.10 Perimeter and area

Lesson
Practice
Lesson

Are you ready?

Let's review how to  find the perimeter  and how to  find the area  of the rectangle.

Examples

Example 1

Find the perimeter of the rectangle shown.

A rectangle with a length of 23 metres and a width of 9 metres.
Worked Solution
Create a strategy

To find the perimeter add the lengths of all the sides of the shape.

Apply the idea
\displaystyle \text{Perimeter}\displaystyle =\displaystyle 9+9+23+23Add the side lengths
\displaystyle =\displaystyle 2 \times 9 +2 \times 23Write as multiplications
\displaystyle =\displaystyle 18 + 46Double 9 and 23
\displaystyle =\displaystyle 64\,\text{m}Add the numbers

Example 2

Find the area of the rectangle shown.

A rectangle with a length of 12 centimetres and a width of 2 centimetres.
Worked Solution
Create a strategy

Use the area of a rectangle formula: \text{Area}=\text{Length} \times \text{Width}

Apply the idea

We can see that length is 12 \text{ cm} and the width is 2 \text{ cm}.

\displaystyle \text{Area}\displaystyle =\displaystyle \text{Length} \times \text{Width}Use the formula
\displaystyle =\displaystyle 12 \times 2 Substitute the length and width
\displaystyle =\displaystyle 24 \text{ cm}^2Double 12

Perimeter and area

You may have noticed already that shapes with the same perimeter don't always have the same area, as shown in the rectangles below. Similarly, shapes with the same area don't always have the same perimeter.

2 rectangles. One with a width of 1 and a length of 4. The other with a width of 2 and a length of 3.

These two shapes have the same perimeter but not the same area.

This video looks at the relationship between perimeter and area.

Loading video...

Examples

Example 3

Which one of these rectangles has an area of 24\text{ cm}^2 and a perimeter of 28\text{ cm}?

A
A rectangle with a width of 2 centimetres and a length of 12 centimetres.
B
A rectangle with a width of 4 centimetres and a length of 10 centimetres.
C
A rectangle with a width of 4 centimetres and a length of 5 centimetres.
D
A rectangle with a width of 3 centimetres and a length of 8 centimetres.
Worked Solution
Create a strategy

Find the area and perimeter of each of the rectangles using the rules:

\text{Area}=\text{length} \times \text{width}

\text{Perimeter}=\text{Sum of all the sides}

Apply the idea

Let's find the areas of each rectangle first.

Option A:

\displaystyle \text{Area}\displaystyle =\displaystyle \text{Length} \times \text{Width}Use the formula
\displaystyle =\displaystyle 12 \times 2Multiply the length and width
\displaystyle =\displaystyle 24\, \text{cm}^2

Option B:

\displaystyle \text{Area}\displaystyle =\displaystyle 10 \times 4Multiply the length and width
\displaystyle =\displaystyle 40\, \text{cm}^2

Option C:

\displaystyle \text{Area}\displaystyle =\displaystyle 5 \times 4Multiply the length and width
\displaystyle =\displaystyle 20\, \text{cm}^2

Option D:

\displaystyle \text{Area}\displaystyle =\displaystyle 8 \times 3Multiply the length and width
\displaystyle =\displaystyle 24\, \text{cm}^2

The areas of options A and D are 24\text{ cm}^2. So now we will find their perimeters.

Option A:

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 12+12+2+2Add the sides
\displaystyle =\displaystyle 28\text{ cm}

Option D:

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 8+8+3+3Add the sides
\displaystyle =\displaystyle 22\text{ cm}

Option A has the correct perimeter. The answer is option A.

Idea summary

Rectangles can have the same perimeter but different areas.

Outcomes

MA3-9MG

selects and uses the appropriate unit and device to measure lengths and distances, calculates perimeters, and converts between units of length

MA3-10MG

selects and uses the appropriate unit to calculate areas, including areas of squares, rectangles and triangles

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