10.02 Perimeter

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.

Exploration

Suppose a farmer wants to make a rectangular paddock for his sheep. He has decided that the paddock must be $20$20 m long and $15$15 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.

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

Perimeter	$=$=	$20+20+15+15$20+20+15+15
	$=$=	$70$70

So the farmer will need $70$70 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.

As we can see from the image, the perimeter of a rectangle will always be:

Perimeter $=$= Length $+$+ Width $+$+ Length $+$+ Width

This is the same as two groups of $($(Length $+$+ Width$)$), which we can write as:

Perimeter $=$=$2$2$\times$×$($(Length $+$+ 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.

As we can see from the image, the perimeter of a square will always be:

Perimeter $=$= Length $+$+ Length $+$+ Length $+$+ Length

This is the same as four groups of the Length, which we can write as:

Perimeter $=$=$4$4$\times$×Length

Let's see these formulae in action.

Worked examples 

example 1

A rectangular race track has a length of $42$42 m and a width of $8$8 m. How long is one lap?

Think: One lap of the race track is equal to its perimeter. Since the race track is a rectangle, we can use the formula,

Perimeter $=$=$2$2$\times$×$($(Length $+$+ Width$)$).

Do: We can find the perimeter of the rectangle by substituting the dimensions of the track into the formula. This gives us:

Perimeter	$=$=	$2\times\left(42+8\right)$2×(42+8)	(Substitute the length and width of the track)
	$=$=	$2\times50$2×50	(Sum the numbers in the brackets)
	$=$=	$100$100	(Perform the multiplication)

So we find that one lap of the race track is $100$100 m long.

Reflect: First we identified that one lap was equal to the perimeter of the race track, then we applied the formula for the perimeter of a rectangle to find the lap length.

example 2

A $32$32 cm long piece of wire is bent into the shape of a square. What is the side length of the square?

Think: Since the piece of wire was $32$32 cm long, we know that the perimeter of the square will be $32$32 cm. We can try reversing the formula for the perimeter of a square, Perimeter $=$=$4$4$\times$×Length, to find a solution.

Do: We know that $32$32 cm $=4\times$=4×Length, so we can find the number that multiplies with $4$4 to give $32$32. This can be found by dividing the perimeter by the number of sides, which gives $\frac{32}{4}=8$324​=8.

So we find that the side length of the square is $8$8 cm.

Reflect: First we identified that the square had a perimeter equal to the length of the wire, then we reversed the formula for the perimeter of a square to find the side 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.

Since it has six sides of equal length we can write its perimeter as:

Perimeter $=$= Length $+$+ Length $+$+ Length $+$+ Length $+$+ Length $+$+ Length

Or more simply:

Perimeter $=$=$6$6$\times$×Length

In fact, we can do the same thing for any regular polygon.

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.

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

Perimeter	$=$=	$2\times\left(8+13\right)$2×(8+13)
	$=$=	$2\times21$2×21
	$=$=	$42$42

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:

Perimeter	$=$=	$2\times\left(5+11\right)+2+2$2×(5+11)+2+2
	$=$=	$2\times16+4$2×16+4
	$=$=	$32+4$32+4
	$=$=	$36$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.

Practice questions

Question 1

Question 2

Question 3