15.04 Composite shapes

When finding the perimeter of composite shapes there are two main approaches.

The first approach is finding the length of all the sides and adding them together like we would for an irregular shape. We can do this by using the lengths we are given to find any missing lengths.

The other approach is less obvious and relies on some visualisation. We can see in the image below that the composite shape actually has the same perimeter as a rectangle.

So the perimeter of this composite shape can be calculated as:

When using this method it is important to keep track of any sides that do not get moved.

An example of a shape that we need to be careful with is:

Notice that we moved the indented edge to complete the rectangle but we still need to count the two edges that weren't moved.

We can calculate the perimeter of this shape as:

With our knowledge of the perimeter of simple shapes like rectangles and squares we can often find creative ways to work out the perimeter of more complicated composite shapes.

Examples

Example 1

Consider the composite shape.

a

Which basic shapes make up this composite shape?

A

Two rhombuses

B

One rhombus

C

Two trapezium

D

One trapezium minus one triangle

Worked Solution

Create a strategy

We can construct a line on the composite shape to break it up into its components.

Apply the idea

As we can see there are two rhombuses in this construction.

So, the correct option is A.

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b

Find the perimeter of the composite shape.

Worked Solution

Create a strategy

To find the perimeter of the composite shape, we want to add all the sides of the shape together.

Apply the idea

Each unlabelled side has the same marking as the side with a length of 11\,cm, so each side has a length of 11\,cm.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 11\times 6	Multiply the length by the number of sides
	\displaystyle =	\displaystyle 66\text{ cm}	Evaluate the product

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Example 2

Consider the composite shape.

a

Which basic shapes make up this composite shape?

A

Three semicircles and one triangle

B

Three quarter circles and one triangle

C

Three semicircles and one square

D

Three quarter circles and one square

Worked Solution

Create a strategy

We can construct some lines on the composite shape to break it up into its components.

Apply the idea

As we can see there are three quarter circles and one square in this construction.

The correct option is D.

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b

Find the exact perimeter of the composite shape.

Worked Solution

Create a strategy

To find the perimeter add the straight sides and the arc lengths of the quarter circles.

Apply the idea

We can find the arc length of one quarter circle by multiplying the circumference formula C=2\pi r by \dfrac{1}{4}. We can find the radius of the quarter circles by dividing the base of the shape by 3 since the length of the square is equal to the radius.

\displaystyle \text{Arc length}	\displaystyle =	\displaystyle \dfrac{1}{4} \times 2\pi r	Multiply the circumference by \dfrac{1}{4}
	\displaystyle =	\displaystyle \dfrac{1}{4} \times 2\pi \times \dfrac{12}{3}	Substitute the radius
	\displaystyle =	\displaystyle 2 \pi	Simplify

To find the perimeter we must add the base, 3 arc lengths and one radius at the top.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 3 \times 2\pi +12 +4	Add all the sides
	\displaystyle =	\displaystyle 6\pi +16\text{ cm}	Simplify

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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.

We may also need to use a combination of the above methods. We can also try and re-arrange or visualise the shape in a different way.

How could we re-visualise the following shape up to make our calculations easier?

The shape consists of a square with side lengths of 6\, \text{cm} and two semi circles.

To find the area, we could work out the area of each semi circle individually, or we could join them back together to make one complete circle. This way we only need to work out the area of one square and one circle. Notice that if we calculated the semi-circular areas separately we are actually halving the area of the circle and then adding the two halves back together.

Similarly, to find the perimeter, by putting the two semi-circles back together we can work out the circumference of the full circle, and then add the two sides of the square that are on the outside of the shape. It is very important that you don't accidentally double-count sides.

Examples

Example 3

Consider the composite shape.

a

Which basic shapes make up this composite shape?

A

A rectangle minus two triangle

B

One rectangle and two trapezium

C

Two parallelograms

D

Two trapeziums

Worked Solution

Create a strategy

We can construct a line on the composite shape to break it up into its components.

Apply the idea

As we can see there are two trapeziums in this construction.

So, the correct option is D.

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b

Find the area of the composite shape.

Worked Solution

Create a strategy

We can find the area of the composite shape by adding the areas of the two trapezium using formula \text{A}=\frac{1}{2}\left(a+b\right)h.

Apply the idea

Since the two trapeziums are identical we can multiply the trapezium formula by 2.

\displaystyle \text{Area}	\displaystyle =	\displaystyle \frac{1}{2}\left(a+b\right)h \times 2	Use the formula
	\displaystyle =	\displaystyle \frac{1}{2}\left(7+15\right)\times 5\times 2	Substitute the values
	\displaystyle =	\displaystyle 110\, \text{cm}^2	Evaluate

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Example 4

Find the area of the composite shape rounded to two decimal places.

Worked Solution

Create a strategy

To find the area of this shape, subtract the area of the semicircle from the area of the trapezium.

Apply the idea

\displaystyle \text{Trapezium area}	\displaystyle =	\displaystyle \dfrac{1}{2}(a+b)h	Use the formula
	\displaystyle =	\displaystyle \frac{1}{2}\left(6+12\right)\times 5	Substitute the values
	\displaystyle =	\displaystyle 45	Evaluate

\displaystyle \text{Semicircle}	\displaystyle =	\displaystyle \dfrac{1}{2} \pi r^2	Use the formula
	\displaystyle =	\displaystyle \dfrac{1}{2} \pi \times \left(\dfrac{6}{2}\right)^2	Substitute the radius
	\displaystyle =	\displaystyle \frac{9\pi }{2}	Simplify

\displaystyle \text{Area}	\displaystyle =	\displaystyle 45-\frac{9\pi }{2}	Subtract the areas
	\displaystyle =	\displaystyle 30.86\, \text{cm}^2	Evaluate and round the answer

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