NSW Mathematics Standard 11 - 2020 Edition
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4.07 Area of composite shapes
Lesson

We are already familiar with a range of basic shapes: squares, rectangles, triangles, parallelograms, trapeziums, circles and sectors.

A composite shape is any shape made up of more than one of the basic shapes.

 

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. 

In the following applet, we explore how a composite shape is made up of smaller basic shapes, and how the areas of the basic shapes can be added together to find the area of the composite shape.

Remember!

There can be more than one way to divide a composite shape into the basic shapes that form it.

 

Practice questions

Question 1

Consider the given shape.

  1. Determine the area of rectangle $B$B.

  2. Hence calculate the total area of the composite shape.

Question 2

Find the area of the shaded region in the figure shown.

Question 3

Consider the shaded area in the adjacent figure (all measurements are in cm). We can find this area by combining the areas of more simple shapes.

  1. First, let's find the area of this big rectangle below.

  2. Next, let's find the area of this coloured rectangle below.

  3. Now find the area of this coloured rectangle below.

  4. Using the answers from the previous parts, find the area of the shaded region in the original figure.

Question 4

Find the area of the shaded region in the following figure, correct to 1 decimal place.

 

Looking for patterns

We can often look for patterns or relationships that will simplify the calculations that we have to perform. This could be the same shape occurring more than once or parts of a shape that add up to a whole.

Let's look at the next example to see how this can work.

 

Worked example

example 1

Find the area of the following composite shape.

Think: We can see straight away that there are four quadrants of a circle and a cross made up of smaller rectangles.

Do: We can separate the shape into a circle and into a cross that is made up of rectangles, work out the area of each part, then add them together to make the whole.

Because they all have the same radius, we can simplify the calculations by finding the area of a full circle, instead of calculating the four quadrants individually.

Area of circle $=$= $\pi r^2$πr2

 

  $=$= $\pi\times5^2$π×52

(Substitute $r=5$r=5)

  $=$= $\pi\times25$π×25

 

  $=$= $78.5398$78.5398...

 

  $=$= $78.5$78.5 units2

 

 

We can then calculate the area of the cross by splitting it into smaller rectangles. There are different ways this could be done. One way is to divide the cross into one rectangle with length $50$50 and width $9$9, and two smaller rectangles of length $18$18 and width $10$10.

Area of cross $=$= $\text{area of large rectangle }+\left(2\times\text{area of small rectangle }\right)$area of large rectangle +(2×area of small rectangle )
  $=$= $\left(50\times9\right)+2\times\left(18\times10\right)$(50×9)+2×(18×10)
  $=$= $450+2\times180$450+2×180
  $=$= $450+360$450+360
  $=$= $810$810 units2

 

Adding the two results together gives the total area of our composite shape.

Total area $=$= $\text{area of circle }+\text{area of cross }$area of circle +area of cross
  $=$= $78.5+810$78.5+810
  $=$= $888.5$888.5 units2

 

Practice questions

Question 5

Find the area of the shape below.

Give your answer correct to 1 decimal place.

Question 6

Find the area of the shaded region in the following figure, correct to 1 decimal place.

Outcomes

MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures

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