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).
A composite shape composed of a large green-shaded rectangle with three smaller rectangular cutouts: one in the center and two in the bottom corners. The top side of the large rectangle is labeled $15$15. The left side of the large rectangle is labeled $7$7. In the center of the large rectangle, a vertically-oriented rectangle is cut out. The top side of this central rectangle is labeled $3$3. The right side of the central rectangle is labeled $11$11. Two identical rectangles are cut out from the bottom left and bottom right corners. The top sides of the cutout identical rectangles are marked with single tick marks. The top side of the left cutout is labeled $4$4. The left side of the of the right cutout is labeled $5$5. The cutout rectangles are unshaded.
First, let's find the area of the entire rectangle shaded below.
A large green-shaded rectangle with three rectangles outlined with dashed lines inside it. The top side of the large rectangle is labeled $15$15. The left vertical side is divided into two segments. The upper segment is labeled $7$7. The lower segment is labeled $5$5. Inside the rectangle, a vertically-oriented rectangle is outlined with dashed lines near the center top. Two additional rectangles are outlined with dashed lines at the bottom left and bottom right corners.
Next, find the area of the interior rectangle shaded below.
A composite shape outlined with dashed green lines, consisting of a large rectangle with two rectangular cutouts at the bottom corners.
Inside the composite shape, a smaller green-shaded rectangle is positioned vertically near the center. The top side of the shaded rectangle is labeled $3$3. The right side of the shaded rectangle is labeled $11$11.Now find the area of the corner rectangle shaded below.
A composite shape outlined with dashed green lines, formed by a large rectangle with a rectangular cutout at the bottom left corner. Inside the composite shape, at the bottom right, a smaller rectangle is shaded in green. The top side of the shaded rectangle is labeled $4$4. The right side of the shaded rectangle is labeled $5$5. In the center of the composite figure, another vertically-oriented rectangle is outlined with green dashed lines.
Using the answers from the previous parts, find the area of the shaded region in the original figure.
A composite shape composed of a large green-shaded rectangle with three smaller rectangular cutouts: one in the center and two in the bottom corners. The top side of the large rectangle is labeled $15$15. The left side of the large rectangle is labeled $7$7. In the center of the large rectangle, a vertically-oriented rectangle is cut out. The top side of this central rectangle is labeled $3$3. The right side of the central rectangle is labeled $11$11. Two identical rectangles are cut out from the bottom left and bottom right corners. The top sides of the cutout identical rectangles are marked with single tick marks. The top side of the left cutout is labeled $4$4. The left side of the of the right cutout is labeled $5$5. The cutout rectangles are unshaded.
Question 3
Find the area of the shaded region in the following figure, correct to one decimal place.
