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Stage 5.1-3

7.01 Composite shapes

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Introduction

Composite shapes made by putting together or taking away pieces of other basic shapes. This concept can be extended into three dimensions, with composite solids being made by adding or subtracting basic solids.

Area of composite shapes

There are two main methods for finding the area of a composite shape: the addition method and the subtraction method

The image shows a composite shape made up of two semi-circles and a trapezium.

The addition method involves breaking the composite shape up into a collection of simple pieces. After finding the areas of each piece, we add them all together to find the total area.

This composite shape is made up of two semi circles and a trapezium.

The image shows a composite shape made up of a trapezium with a semi circle cut out of it at the top.

The subtraction method thinks of the composite shape as a larger shape with smaller pieces taken away from it. We find the area of the small shapes and take that area away from the large shape to find the total area.

These two methods are very useful - sometimes we could use either, and sometimes we should use both.

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...

By finding the area of each simple shape, we can add them to find the area of the composite shape.

Examples

Example 1

Find the total area of the figure shown.

A composite shape made up of a rectangle, triangle and parallelogram. Ask your teacher for more information.
Worked Solution
Create a strategy

Add the area of the simple shapes that make up the composite shape.

Apply the idea

The composite shape is made up of a parallelogram, a triangle, and a rectangle so we need to find the area of each and add them.

\displaystyle \text{Area of parallelogram}\displaystyle =\displaystyle bhUse the area formula
\displaystyle =\displaystyle 24 \times 32Substitute the base and height
\displaystyle =\displaystyle 768\text{ mm}^{2}Evaluate the multiplication
\displaystyle \text{Area of triangle}\displaystyle =\displaystyle \dfrac{1}{2}bhUse the area formula
\displaystyle =\displaystyle \dfrac{1}{2} \times 16 \times 32Substitute the base and height
\displaystyle =\displaystyle 256\text{ mm}^{2}Evaluate the multiplication
\displaystyle \text{Area of rectangle}\displaystyle =\displaystyle lwUse the area formula
\displaystyle =\displaystyle 56 \times 18Substitute the height and width
\displaystyle =\displaystyle 1008\text{ mm}^{2}Evaluate the multiplication
\displaystyle \text{Total Area}\displaystyle =\displaystyle 768 + 256 + 1008Add the areas
\displaystyle =\displaystyle 2032\text{ mm}^{2}Evaluate

Example 2

Consider the composite shape.

A composite shape made up of a trapezium, rectangle with quarter circles cut out. Ask your teacher for more information.
a

Which basic shapes make up this composite shape?

A
One trapezium and one rectangle minus two semicircles.
B
One trapezium and one rectangle minus two quarter circles.
C
One trapezium and one parallelogram minus two quarter circles.
D
One trapezium and one rectangle.
Worked Solution
Create a strategy

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

Apply the idea
A composite shape made up of a trapezium, rectangle with quarter circles cut out. Ask your teacher for more information.

By drawing lines we can see a trapezium at the top, and a rectangle at the bottom with two quarter circles cut out from each side.

The correct answer is option B: One trapezium and one rectangle minus two quarter circles.

b

Find the area of the composite shape. Round your answer to two decimal places.

Worked Solution
Create a strategy

We need to add the areas of the trapezium and the rectangle together, then take away the areas of the two quarter circles.

Apply the idea
\displaystyle \text{Trapezium area}\displaystyle =\displaystyle \dfrac{1}{2}\,(a + b)hUse the area formula
\displaystyle =\displaystyle \dfrac{1}{2}\,(7 + 12) \times 6Substitute a, b and h
\displaystyle =\displaystyle 57\text{ cm}^{2}Evaluate
\displaystyle \text{Rectangle area}\displaystyle =\displaystyle lwUse the area formula
\displaystyle =\displaystyle 12\times 4Substitute l,w
\displaystyle =\displaystyle 48\text{ cm}^{2} Evaluate

The area of the 2 quarter circle is the same as the area of 1 semi circle.

\displaystyle \text{Semi circle area}\displaystyle =\displaystyle \dfrac{1}{2} \pi r^{2}Use the area formula
\displaystyle =\displaystyle \dfrac{1}{2}\times \pi \times 4^{2}Substitute r
\displaystyle =\displaystyle 25.13\text{ cm}^{2}Evaluate and round
\displaystyle \text{Composite area}\displaystyle =\displaystyle 57 + 48 - 25.13Combine the areas
\displaystyle =\displaystyle 79.87\text{ cm}^{2}Evaluate
Idea summary

There are two main methods for finding the area of a composite shape: the addition method and the subtraction method:

  • The addition method involves breaking the composite shape up into simple shapes and adding the areas together.

  • The subtraction method involves finding the area of the larger shape and subtracting the smaller areas that were cut out.

Volume of composite solids

To find the volume of a composite solid, we use the same addition and subtraction methods - but this time we use basic solids rather than basic shapes.

If a prism has a complicated shape for a base, it is often easiest to find the composite area of the base as usual, using the addition or subtraction method. Then we can use the volume formula for a prism V=Ah, where A is its base area and h is its perpendicular height.

Examples

Example 3

Calculate the volume of the solid correct to one decimal place.

A composite solid made of a rectangular prism and half a cylindrical prism on top. Ask your teacher for more information.
Worked Solution
Create a strategy

We can find the volume of the rectangular prism and add it to the volume of half the cylinder.

Apply the idea
\displaystyle \text{Rectangular prism volume}\displaystyle =\displaystyle lwhUse the volume formula
\displaystyle =\displaystyle 6 \times 14 \times 5Substitute w, \,h, and l
\displaystyle =\displaystyle 420 \text{ cm}^{3}Evaluate

To find the volume of the half cylinder:

\displaystyle \text{Half cylinder volume}\displaystyle =\displaystyle \dfrac{1}{2} \pi r^{2} hUse the volume formula
\displaystyle =\displaystyle \dfrac{1}{2} \times \pi \times \left(\dfrac{6}{2}\right)^{2} \times 14Substitute r, \,h
\displaystyle =\displaystyle 197.9 \text{ cm}^{3}Evaluate
\displaystyle V\displaystyle =\displaystyle 420 + 197.9Add the volumes
\displaystyle =\displaystyle 617.9\text{ cm}^{3}Evaluate

Example 4

Find the volume of this composite solid, created by passing a cylinder through a cube.

A cube with cylindrical hole. The side length of the cube is 72 millimetres. The radius of the cylinder is 36 millimetres.

Give your answer correct to two decimal places.

Worked Solution
Create a strategy

We can find the volume of the cube and subtract the volume of the cylinder.

Apply the idea

To find the volume of cube:

\displaystyle \text{Cube volume}\displaystyle =\displaystyle s^{3}Use the volume formula
\displaystyle =\displaystyle 72^{3}Substitute s
\displaystyle =\displaystyle 373\,248 \text{ mm}^{3}Evaluate
\displaystyle \text{Cylinder volume}\displaystyle =\displaystyle \pi r^{2} hUse the volume formula
\displaystyle =\displaystyle \pi \times 36^{2} \times 72Substitute r, \, h
\displaystyle =\displaystyle 293\,148.29 \text{ mm}^{3}Evaluate and round
\displaystyle V\displaystyle =\displaystyle 373\,248 - 293\,148.29Subtract the volumes
\displaystyle =\displaystyle 80\,099.71\text{ mm}^{3}Evaluate
Idea summary

If a prism has a complicated shape for a base, it is often easiest to find the composite area of the base using the addition or subtraction method. Then to find the volume we use the formula V=Ah.

Outcomes

MA5.1-8MG

calculates the areas of composite shapes, and the surface areas of rectangular and triangular prisms

MA5.2-11MG

calculates the surface areas of right prisms, cylinders and related composite solids

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