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 triangle at the top of the shape is:
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 at the bottom of the shape is:
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 figure shown below (all measurements are in cm).
First, let's find the area of the entire rectangle shaded below.
A large rectangle and inside it are three smaller interior rectangles outlined with dashed lines, indicating areas that will be cut out. The three dashed rectangles are located in the lower-left corner, in the middle, and in the lower-right corner. The length of the large rectangle is $15$15 units as labeled. Its width is divided and labeled in two segments, $5$5 units and $7$7 units. Next, find the area of the interior rectangle shaded below.
A composite figure outline in dashed green line and within it is an interior rectangle. The interior rectangle is shaded in green and has a length of $3$3 units and a width of $11$11 units as labeled. Now find the area of the corner rectangle shaded below.
A composite figure outline in dashed green line and within it are two rectangles: one in the middle and outlined in dashed green line, and another at the lower-right corner and shaded in green. The green-shaded corner rectangle has a length of $4$4 units and a width of $5$5 units. Using the answers from the previous parts, find the area of the shaded region in the original figure.
A large rectangle with three smaller interior rectangles cut out from it resulting in a composite figure shaded in green. The three smaller interior rectangles are outlined in green dashed lines, and are located in the lower-left corner, in the middle, and in the lower-right corner. The dimensions are labeled. The large rectangle has a length of $15$15 units and a width of $12$12 units. The lower-left and lower-right rectangles are identical, each with a length of $4$4 units and a width of $5$5 units. The middle rectangle has a length of $3$3 units and a width of $11$11 units.
Question 3
Find the area of the shaded region in the following figure, correct to one decimal place.
