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Year 8

8.05 Composite shapes

Lesson

Introduction

A composite shape is one that is made from a number of smaller shapes. We have looked at shapes made up of rectangles and triangles and in this chapter we will extend the idea to include the shapes we have just looked at: rhombuses, parallelograms, trapeziums, kites and circles.

We can use the properties of these regular shapes to to learn more about the composite shape. For example, knowing the total area of all the smaller shapes is the same as knowing the area of the whole composite shape.

A composite shape. Ask your teacher for more information.

Dashed lines can be used to visualise which simple shapes make up a composite shape.

Perimeter of 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.

A composite shape transformed into rectangle with length of 13 and width of 8. Ask your teacher for more information.

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

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2 \times (8+13)
\displaystyle =\displaystyle 2 \times 21
\displaystyle =\displaystyle 42

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:

A composite shape with measurements of 5, 2, and 11. Ask your teacher for more information.

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:

\displaystyle \text{Perimeter}\displaystyle =\displaystyle 2 \times (5+11) +2 +2
\displaystyle =\displaystyle 2 \times 16+4
\displaystyle =\displaystyle 32+4
\displaystyle =\displaystyle 36

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 6 sided composite shape with a side length of 11 centimetres. Ask your teacher for more information.
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
A 6 sided composite shape with a dashed line down the middle. Ask your teacher for more information

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

So, the correct option is A.

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

Example 2

Consider the composite shape.

A composite shape with a base of 12 centimetres. Ask your teacher for more information
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
A composite shape broken up into a square and 4 quarter circles. Ask your teacher for more information.

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

The correct option is D.

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 rMultiply the circumference by \dfrac{1}{4}
\displaystyle =\displaystyle \dfrac{1}{4} \times 2\pi \times \dfrac{12}{3}Substitute the radius
\displaystyle =\displaystyle 2 \piSimplify

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 +4Add all the sides
\displaystyle =\displaystyle 6\pi +16\text{ cm}Simplify
Idea summary

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 by mapping sides to form a familiar shape such as a rectangle.

Area of composite shapes

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?

A shape consists of a square with side lengths of 6 centimetres and two semi circles. Ask your teacher for more information

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.

Exploration

The following applet shows how a composite shape can be broken down into pieces of basic shapes in order to find the area.

Loading interactive...

We can find the area of a composite shape by dividing it into simple shapes, and finding the area of each simple shape one at a time. Then we can add these areas to find the total area of the composite shape.

Examples

Example 3

Consider the composite shape.

 A composite shape with 6 sides. Ask your teacher for more information
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
A composite shape with 6 sides divided into 2 trapeziums. Ask your teacher for more information

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

So, the correct option is D.

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

Example 4

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

A composite shape with a semi circle cut out of a trapezium. Ask your teacher for more information.
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)hUse the formula
\displaystyle =\displaystyle \frac{1}{2}\left(6+12\right)\times 5Substitute the values
\displaystyle =\displaystyle 45Evaluate
\displaystyle \text{Semicircle}\displaystyle =\displaystyle \dfrac{1}{2} \pi r^2Use the formula
\displaystyle =\displaystyle \dfrac{1}{2} \pi \times \left(\dfrac{6}{2}\right)^2Substitute 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}^2Evaluate and round the answer
Idea summary

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.

Outcomes

ACMMG196

Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites

ACMMG197

Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving circumference and area

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