Area of composite shapes
A composite shape is a shape that can be broken up into smaller more recognisable shapes.
For example, this shape is a square and a triangle combined.
Finding the are of a composite shape requires us to be able to break up the shape into simpler parts.
For the picture drawn, we don't immediately know how to find the area, but we do know how to find the area of a square and a triangle.
Cutting up shapes
Now sometimes it's easier to see small shapes that make up a large one. Like this one. We could break it up into smaller rectangles.
Sometimes it's easier to see a large shape with a bit missing. It's good to get practice at both. So for the same shape we might see a large rectangle with a small rectangle cut out of it.
Worked examples
example 1
Find the area of the composite shape below.
Think: What shapes can we break this up into? In this case, it will be easiest to split it into two triangles.
Do: The first triangle I see is this one.
The base measurement is $3-1=2$3−1=2 cm
The height measurement is $3.2$3.2 cm
| $\text{Area of a triangle }$Area of a triangle | $=$= | $\frac{1}{2}\times b\times h$12×b×h |
| $=$= | $\frac{1}{2}\times2\times3.2$12×2×3.2 | |
| $=$= | $3.2$3.2 cm2 |
The other triangle is this one:
The base measurement is $3$3 cm.
The height measurement is $3$3 cm.
| $\text{Area of a triangle }$Area of a triangle | $=$= | $\frac{1}{2}\times b\times h$12×b×h |
| $=$= | $\frac{1}{2}\times3\times3$12×3×3 | |
| $=$= | $4.5$4.5 cm2 |
So the total area of the composite shape is $3.2+4.5=7.7$3.2+4.5=7.7 cm2
example 2
A backyard garden needs to have turf laid. The shape and dimensions of the garden are indicated in the picture below. Find the area of the turf required.
Think: What shapes could we break this up into? This time, let's use a large rectangle and cut off a triangle from the corner.
So the total area will be the area of shape 1 (the rectangle) minus the area of shape 2 (the triangle).
Do:
The outside rectangle has:
| $\text{Area }$Area | $=$= | $L\times W$L×W |
| $=$= | $12\times9$12×9 | |
| $=$= | $108$108 m2 |
The corner triangle has
| $\text{Area }$Area | $=$= | $\frac{1}{2}\times b\times h$12×b×h |
| $=$= | $\frac{1}{2}\times\left(12-5\right)\times\left(9-4\right)$12×(12−5)×(9−4) | |
| $=$= | $\frac{1}{2}\times7\times5$12×7×5 | |
| $=$= | $17.5$17.5 m2 |
So the total area is the area of the rectangle minus the area of the triangle
| $\text{Total Area}$Total Area | $=$= | $108-17.5$108−17.5 |
| $=$= | $90.5$90.5 m2 |
Practice questions
Question 1
Find the area of the figure shown.
Question 2
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.
First, let's find the area of this big rectangle below.
Next, let's find the area of this coloured rectangle below.
Now find the area of this coloured rectangle below.
Using the answers from the previous parts, find the area of the shaded region in the original figure.
Question 3
Find the area of the shaded region in the following figure, correct to one decimal place.
