10.02 Perimeter

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

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

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.

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 10	Substitute the length
	\displaystyle =	\displaystyle 40 \text{ cm}	Evaluate

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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.

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?

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

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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.

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

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:

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:

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.

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+4	Add the sides
	\displaystyle =	\displaystyle 48 \text{ cm}	Evaluate

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Example 5

Consider the following figure.

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

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

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

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