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14.02 Perimeter

Lesson

Introduction

The perimeter of a shape is the length of its outline. For example, the perimeter of a rectangle is the combined length of all its four sides.

Perimeters are useful for when we know the dimensions of a shape but not the distance around it, or maybe we know the distance around it and want to find its dimensions.

Suppose a farmer wants to make a rectangular paddock for his sheep. He has decided that the paddock must be 20\text{ m} long and 15\text{ m} wide, and needs to determine how much fencing to purchase.

The farmer can work out the required length of fencing by finding the perimeter of the rectangle enclosed by the fence.

 A rectangular paddock for sheep with length equal to 20 metres and width equal to 15 metres.

We can see that the rectangular paddock will have two sides of length 20\text{ m} and two sides of length 15\text{ m.} Adding these four sides together will give us the perimeter.

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 20+20+15+15
\displaystyle =\displaystyle 70

So the farmer will need 70\text{ m} of fencing to make his paddock.

Perimeter of a rectangle

Every rectangle has two pairs of equal sides, which we can call the length and the width

A rectangle showing its labels of length and width.

As we can see from the image, the perimeter of a rectangle will always be:\text{Perimeter} = \text{Length} + \text{Width} + \text{Length} + \text{Width}

This is the same as two groups of \text{(Length} + \text{Width)} which we can write as: \text{Perimeter} = 2 \times \text{(Length} + \text{Width)}

Examples

Example 1

Find the perimeter of a rectangle with a length of 20 cm and a width of 5 cm.

Worked Solution
Create a strategy

Use the formula for the perimeter of the rectangle.

Apply the idea

We are given \text{length}=20 and \text{width}=5.

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2 \times (20+5)Substitute the length and width
\displaystyle =\displaystyle 2 \times 25Find the sum in the brackets
\displaystyle =\displaystyle 50 \text{ cm}Evaluate
Idea summary

The perimeter of a rectangle has the formula: \text{Perimeter} = 2 \times \text{(Length} + \text{Width)}

Perimeter of a square

The main property of a square that we can use to calculate its perimeter is that it has four equal sides.

A square showing its label of side length.

As we can see from the image, the perimeter of a square will always be:\text{Perimeter} = \text{Length} + \text{Length} + \text{Length} + \text{Length}

This is the same as four groups of the Length, which we can write as: \text{Perimeter} = 4 \times \text{ Length}

Examples

Example 2

Find the perimeter of the square shown.

A square with side length of 10 centimetres.
Worked Solution
Create a strategy

Use the formula for the perimeter of the square.

Apply the idea

The side length is \text{Length}=10 cm.

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 4 \times\text{Length}Use the formula
\displaystyle =\displaystyle 4\times 10Substitute the length
\displaystyle =\displaystyle 40 \text{ cm}Evaluate
Idea summary

The perimeter of a square has the formula: \text{Perimeter} = 4 \times \text{ Length}

Perimeter of regular polygons

Regular polygons are special in that all of their sides are the same length. A regular pentagon has five equal sides, a regular octagon has eight equal sides, and so on. This property of regular polygons makes it quite simple to find their perimeter.

Consider a regular hexagon.

A hexagon showing its label of side lengths.

Since it has six sides of equal length we can write its perimeter as:\text{Perimeter} = \text{Length} + \text{Length} + \text{Length} + \text{Length}+ \text{Length}+ \text{Length}Or more simply: \text{Perimeter} = 6 \times \text{ Length} In fact, we can do the same thing for any regular polygon.

Examples

Example 3

This shape has a perimeter of 64 cm. What is the length of each side?

An octagon with perimeter of 64 centrimetres.
Worked Solution
Create a strategy

Divide the perimeter by the number of sides. Use the fact that the shape shown is a regular octagon which has 8 sides of equal length.

Apply the idea
\displaystyle 64\displaystyle =\displaystyle 8 \times\text{Length}Substitute the perimeter
\displaystyle \dfrac{64}{8}\displaystyle =\displaystyle \dfrac{ 8\times \text{Length}}{8}Divide both sides by 8
\displaystyle \text{Length}\displaystyle =\displaystyle 8 \text{ cm}Evaluate
Idea summary

The perimeter of a regular n-gon (a polygon with n-sides) has the formula: \text{Perimeter} = n \times \text{Length}

Perimeter of composite shapes

When finding the perimeter of composite shapes there are two main approaches.

The first approach is finding the length of all the sides and adding them together like we would for an irregular shape. We can do this by using the lengths we are given to find any missing lengths.

The other approach is less obvious and relies on some visualisation. We can see in the image below that the composite shape actually has the same perimeter as a rectangle.

A composite shape transformed into rectangle with length of 13 and width of 8. Ask your teacher for more information.

So the perimeter of this composite shape can be calculated as:

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2 \times (8+13)
\displaystyle =\displaystyle 2 \times 21
\displaystyle =\displaystyle 42

When using this method it is important to keep track of any sides that do not get moved.

An example of a shape that we need to be careful with is:

A composite shape with measurements of 5, 2, and 11. Ask your teacher for more information.

Notice that we moved the indented edge to complete the rectangle but we still need to count the two edges that weren't moved.

We can calculate the perimeter of this shape as:

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2 \times (5+11) +2 +2
\displaystyle =\displaystyle 2 \times 16+4
\displaystyle =\displaystyle 32+4
\displaystyle =\displaystyle 36

With our knowledge of the perimeter of simple shapes like rectangles and squares we can often find creative ways to work out the perimeter of more complicated composite shapes.

Examples

Example 4

Find the perimeter of the shape given.

A composite shape with 5 different side lengths in centimetres. Ask your teacher for more information.
Worked Solution
Create a strategy

The perimeter of a figure is the sum of all of its sides.

Apply the idea
\displaystyle \text{Perimeter}\displaystyle =\displaystyle 8+8+16+12+4Add the sides
\displaystyle =\displaystyle 48 \text{ cm}Evaluate

Example 5

Consider the following figure.

A composite shape with 4 different side lengths in metres and with missing x and y lengths. Ask your teacher for more information.
a

Find the length x.

Worked Solution
Create a strategy

The side length x will be equal to the difference between the other two vertical side lengths. Use the fact that the vertical side on the left has the same length as the two vertical sides on the right added together.

Apply the idea
\displaystyle 5\displaystyle =\displaystyle x+2Write the relationship in an equation
\displaystyle 5-2\displaystyle =\displaystyle x+2-2Subtract 2 on both sides
\displaystyle x\displaystyle =\displaystyle 3 \text{ m}Evaluate
b

Find the length y.

Worked Solution
Create a strategy

The side length y will be equal to the difference between the other two horizontal side lengths. Use the fact that the horizontal side on the bottom has the same length as the two horizontal sides on the top added together.

Apply the idea
\displaystyle 17\displaystyle =\displaystyle y+7Write the relationship in an equation
\displaystyle 17-7\displaystyle =\displaystyle y+7-7Subtract 7 on both sides
\displaystyle y\displaystyle =\displaystyle 10 \text{ m}Evaluate
c

Calculate the perimeter of the figure.

Worked Solution
Create a strategy

The perimeter of a figure is the sum of all of its sides. Use the results found in parts (a) to (b).

Apply the idea
\displaystyle \text{Perimeter}\displaystyle =\displaystyle 7+x+y+2+17+5Add the sides
\displaystyle =\displaystyle 7+3+10+2+17+5Substitute x and y
\displaystyle =\displaystyle 44 \text{ m}Evaluate
Idea summary

When finding the perimeter of composite shapes there are two main approaches.

The first approach is finding the length of all the sides and adding them together like we would for an irregular shape. We can do this by using the lengths we are given to find any missing lengths.

The other approach is less obvious and relies on some visualisation by mapping sides to form a familiar shape such as a rectangle.

Outcomes

MA4-12MG

calculates the perimeters of plane shapes and the circumferences of circles

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