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9.05 Composite shapes

Lesson

A composite shape is one that is made from a number of smaller shapes. We have looked at shapes made up of rectangles and triangles and in this chapter we will extend the idea to include the shapes we have just looked at: rhombuses, parallelograms, trapeziums, kites and circles.

We can use the properties of these regular shapes to to learn more about the composite shape. For example, knowing the total area of all the smaller shapes is the same as knowing the area of the whole composite shape.

Dashed lines can be used to visualise which simple shapes make up a composite shape.

Finding the perimeter of a composite shape

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

The first approach is to find the length of all the sides and add 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
Careful!

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:

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

 

Practice questions

Question 1

Consider the composite shape.

  1. Which basic shapes make up this composite shape?

    Two rhombuses

    A

    One rhombus

    B

    Two trapeziums

    C

    One trapezium minus one triangle

    D

    Two rhombuses

    A

    One rhombus

    B

    Two trapeziums

    C

    One trapezium minus one triangle

    D
  2. Find the perimeter of the composite shape.

Question 2

Consider the composite shape.

  1. Which basic shapes make up this composite shape?

    Three semicircles and one triangle

    A

    Three quarter circles and one triangle

    B

    Three semicircles and one square

    C

    Three quarter circles and one square

    D

    Three semicircles and one triangle

    A

    Three quarter circles and one triangle

    B

    Three semicircles and one square

    C

    Three quarter circles and one square

    D
  2. Find the exact perimeter of the composite shape.

Finding the area of a composite shape

To calculate the area of a composite shape, we can use either of two methods:

  • Addition method - Divide the composite shape into basic shapes, work out the area of each basic shape, then add them together.
  • Subtraction method - Work out the area of the basic shape that encloses the composite shape, then subtract the areas of smaller basic shapes as necessary. 

We may also need to use a combination of the above methods. We can also try and re-arrange or visualise the shape in a different way.

How could we re-visualise the following shape up to make our calculations easier?

The shape consists of a square with side lengths of $10$10 cm and two semi circles

To find the area, we could work out the area of each semi circle individually, or we could join them back together to make one complete circle. This way we only need to work out the area of one square and one circle. Notice that if we calculated the semi-circular areas separately we are actually halving the area of the circle and then adding the two halves back together.

Similarly, to find the perimeter, by putting the two semi-circles back together we can work out the circumference of the full circle, and then add the two sides of the square that are on the outside of the shape. It is very important that you don't accidentally double-count sides.

 

Practice questions

Question 1

Consider the composite shape.

  1. Which basic shapes make up this composite shape?

    A rectangle minus two triangles

    A

    One rectangle and two trapeziums

    B

    Two parallelograms

    C

    Two trapeziums

    D

    A rectangle minus two triangles

    A

    One rectangle and two trapeziums

    B

    Two parallelograms

    C

    Two trapeziums

    D
  2. Find the area of the composite shape.

Question 2

Find the area of the composite shape rounded to two decimal places.

Outcomes

MA4-13MG

uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area

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