10.05 Area of composite shapes

It may be easier to see small shapes that make up a large one ("adding" shapes together).

Sometimes it's easier to see a large shape with a bit missing ("subtracting" shapes). It's good to get practice at both. So, the same shape could be a large rectangle with a small rectangle cut out of it.

What shapes can you see in these pictures? Practice breaking the composite shape up into smaller parts, or looking for a larger shape with a piece cut out of it. Remember to look for shapes already studied such as rectangles, squares, triangles or parallelograms.

To find the areas of composite shapes, being able to identify the shapes is only the first step. The next is calculating the areas of the parts. Consider the following two examples from the shapes above.

Examples

Example 1

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.

Worked Solution

Create a strategy

We need to subtract the area of the corner triangle from the area of the rectangle.

Apply the idea

Area of rectangle:

\displaystyle A	\displaystyle =	\displaystyle l \times w	Write the formula
	\displaystyle =	\displaystyle 12 \times 9	Substitute l=12 and w=9
	\displaystyle =	\displaystyle 108\text{ m}^2	Evaluate

Area of triangle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac{1}{2}bh	Write the formula
	\displaystyle =	\displaystyle \dfrac{1}{2}\times (12-5) \times (9-4)	Substitute b=12-5 and h=9-4
	\displaystyle =	\displaystyle \dfrac{1}{2}\times 7 \times 5	Evaluate the multiplication
	\displaystyle =	\displaystyle 17.5\text{ m}^2	Evaluate

Area of the turf:

\displaystyle A	\displaystyle =	\displaystyle \text{Area of the rectangle - area of the triangle}	Write the formula
	\displaystyle =	\displaystyle 108 - 17.5	Substitute the areas
	\displaystyle =	\displaystyle 90.5\text{ m}^2	Evaluate

Example 2

Find the total area of the figure shown.

Worked Solution

Create a strategy

Add the areas of the two triangles.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \left(\dfrac{1}{2}\times 12 \times 4\right) + \left(\dfrac{1}{2}\times 12 \times 7\right)	Susbtitute b=12,\,h=4 and h=7
	\displaystyle =	\displaystyle 24+42	Evaluate the multiplication in each brackets
	\displaystyle =	\displaystyle 66\text{ m}^2	Evaluate

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Example 3

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

Worked Solution

Create a strategy

Subtract the area of the triangle from the area of the square.

Apply the idea

Area of square:

\displaystyle A	\displaystyle =	\displaystyle s^{2}	Write the formula
	\displaystyle =	\displaystyle 9^{2}	Substitute s=9
	\displaystyle =	\displaystyle 81\text{ cm}^2	Evaluate

Area of triangle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac{1}{2}bh	Write the formula
	\displaystyle =	\displaystyle \dfrac{1}{2}\times 3 \times 3	Substitute b=3 and h=3
	\displaystyle =	\displaystyle 4.5\text{ cm}^2	Evaluate

Area of the shaded region:

\displaystyle A	\displaystyle =	\displaystyle \text{Area of square - Area of triangle }	Write the formula
	\displaystyle =	\displaystyle 81-4.5	Substitute the areas of square and triangle
	\displaystyle =	\displaystyle 76.5\text{ cm}^2	Evaluate

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An annulus is a doughnut shape where the area is formed by two circles with the same centre.

\text{Area of an annulus} = \text{area of larger circle - area of smaller circle}

Examples

Example 4

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

Worked Solution

Create a strategy

Subtract the smaller circle's area from the larger circle.

Apply the idea

The radius of the smaller circle is r=\dfrac{15}{2}=7.5\text{ cm,} and the radius of the larger circle is r=\dfrac{20}{2}=10\text{ cm}.

\displaystyle \text{Area of annulus}	\displaystyle =	\displaystyle \pi \times 10^2-\pi \times 7.5^2	Subtract the areas
	\displaystyle =	\displaystyle 100\pi - 56.25\pi	Simplify
	\displaystyle =	\displaystyle 137.4\, \text{cm}^2	Evaluate

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Area of sectors:

\text{Area of a circle}=\pi r^{2}

\text{Area of a semi-circle}=\dfrac{1}{2}\pi r^{2}

\text{Area of a quarter-circle}=\dfrac{1}{4}\pi r^{2}

Always consider the fraction of the circle. For example consider a quarter circle-the angle of this sector is 90\degree. The fraction of the circle is \dfrac{90}{360}=\dfrac{1}{4}. Therefore to calculate the area of a quarter of a circle, we calculate the whole circle and divide by 4.

Examples

Example 5

Calculate the area of the following figure, correct to one decimal place.

Worked Solution

Create a strategy

We can use the formula: A=\dfrac{3}{4}\pi r^{2}.

Apply the idea

We use \dfrac{3}{4} since the circle is cut out its \dfrac{1}{4}.

\displaystyle A	\displaystyle =	\displaystyle \dfrac{3}{4}\times \pi \times 2^{2}	Substitute r=2
	\displaystyle =	\displaystyle 9.4\text{ cm}^2	Evaluate

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Example 6

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

Worked Solution

Create a strategy

Subtract the area of the quarter circle from the area of the square.

Apply the idea

Area of the quarter-circle:

\displaystyle A	\displaystyle =	\displaystyle \dfrac{1}{4}\pi r^{2}	Write the formula
	\displaystyle =	\displaystyle \dfrac{1}{4}\times \pi \times (20)^{2}	Substitute r=20
	\displaystyle =	\displaystyle 314.2\text{ cm}^2	Evaluate

Area of the square:

\displaystyle A	\displaystyle =	\displaystyle s^{2}	Write the formula
	\displaystyle =	\displaystyle (24)^2	Substitute s=24
	\displaystyle =	\displaystyle 576\text{ cm}^2	Evaluate

Area of the shaded region:

\displaystyle A	\displaystyle =	\displaystyle \text{Area of the square} - \text{Area of the quarter-circle}	Write the formula
	\displaystyle =	\displaystyle 576 - 314.2	Substitute the areas
	\displaystyle =	\displaystyle 261.8\text{ cm}^2	Evaluate

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Example 7

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

Worked Solution

Create a strategy

Subtract the area of the 4 semi-circles from the area of the larger square.

Apply the idea

Area of the 4 semi-circle:

\displaystyle A	\displaystyle =	\displaystyle \pi r^{2} \times \dfrac{1}{2} \times 4	Write the formula
	\displaystyle =	\displaystyle \pi \times \left(10 \times \dfrac{1}{2}\right)^{2}\times \dfrac{1}{2} \times 4	Substitute r=\left(10 \times \dfrac{1}{2}\right)
	\displaystyle =	\displaystyle 157.1\text{ cm}^2	Evaluate

Area of the square:

\displaystyle A	\displaystyle =	\displaystyle s^{2}	Write the formula
	\displaystyle =	\displaystyle (20)^2	Substitute s=20
	\displaystyle =	\displaystyle 400\text{ cm}^2	Evaluate

Area of the shaded region:

\displaystyle A	\displaystyle =	\displaystyle \text{Area of the square} - \text{Area of the semi-circles}	Write the formula
	\displaystyle =	\displaystyle 400 - 157.1	Substitute the areas
	\displaystyle =	\displaystyle 242.9\text{ cm}^2	Evaluate

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