5.03 Perimeter of composite shapes

The perimeter of a 2D shape is the total distance around the boundary of the shape. For polygons (2D shapes made up of straight edges) we can find the perimeter by adding up the lengths of all the edges. This includes triangles, squares, rectangles, trapezia and other irregular polygons like the one below.

Perimeter $=16+12+4+8+8=48$=16+12+4+8+8=48 m

 

In practical problems we often encounter shapes that are combinations of sectors and polygons, as well as shapes with holes in them. These are known as composite shapes.

For these shapes we can find the perimeter by adding together all the lengths of the straight line segments and the arc lengths, remembering that if the shape has a hole in it then there is an internal boundary that also counts towards the perimeter.

                   

We also encounter problems where not all of the side lengths are known. In many applications - such as surveying the dimensions of a block of land - taking length measurements can be time consuming, and so if we can work out some side lengths with a bit of thought, or by using formulas like Pythagoras' theorem, it will save time.

Worked example

Find the perimeter of the shape pictured below, where the measurements are in metres:

Think: At first glance it might look like we don't have enough information to add up all the sides. If we think carefully, however, notice that the two unknown horizontal side lengths must add up to $13$13 m. Similarly, the two unknown vertical side lengths must add up to $8$8 m. We can picture this by moving the sides to the edge as follows:

This shape is a rectangle with exactly the same perimeter as our original shape (although a different area).

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

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

Practice questions

Question 1

Question 2

Question 3