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Stage 5.1-2

9.09 Composite solids

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Introduction

Composite solids are 3D figures comprising multiple simpler solids, either added together or subtracted from one another.

Volume of composite solids

Some composite solids are made by joining simple solids together. We can find the volume of these composite solids by finding the volume of each of the pieces and adding them.

A composite solid made by cutting out half a cylinder from the top a rectangular prism.

In the case where the composite solid is made by subtracting one simple solid from another, as seen here, we can subtract volumes of simple solids to find the volume of the composite solid.

This composite solid is made by subtracting half a cylinder from a rectangular prism.

In the case where the composite solid is also a prism with some composite shape as its base, we can find its volume in the same way that we'd find the volume of a prism.

Examples

Example 1

Find the volume of the figure shown, correct to two decimal places:

A composite solid made of a square prism and half a cylinder. Ask your teacher for more information.
Worked Solution
Create a strategy

Add the volume of the square prism to the volume of the half cylinder.

Apply the idea

The half cylinder has a diameter of 4\text{ m} which means its radius is 2\text{ m}.

\displaystyle \text{Half cylinder volume}\displaystyle =\displaystyle \dfrac{1}{2}\pi r^2 hWrite the formula
\displaystyle =\displaystyle \dfrac{1}{2} \pi \times 2^2 \times 10Substitute r=2,\, h=10
\displaystyle \approx\displaystyle 62.83Evaluate and round
\displaystyle \text{Prism volume}\displaystyle =\displaystyle s^2hWrite the formula
\displaystyle =\displaystyle 4^2 \times 10Substitute s=4,\, h=10
\displaystyle =\displaystyle 160Evaluate
\displaystyle V\displaystyle =\displaystyle 62.83+160Add the volumes
\displaystyle =\displaystyle 222.83\text{ m}^3Evaluate
Reflect and check

Since the composite solid was made by adding simple solids together, its volume was the sum of the volumes of the simple solids.

Example 2

This solid consists of a rectangular prism with a smaller rectangular prism cut out of it. Find the volume of the solid.

A large rectangular prism with a small rectangular prism cut out. Ask your teacher for more information.
Worked Solution
Create a strategy

Subtract the volume of the smaller rectangular prism from the volume of the larger rectangular prism.

Apply the idea
\displaystyle \text{Large prism}\displaystyle =\displaystyle 6\times 15 \times 13Multiply the dimensions
\displaystyle =\displaystyle 1170\text{ cm}^3Evaluate
\displaystyle \text{Small prism}\displaystyle =\displaystyle 11\times 2 \times 15Multiply the dimensions
\displaystyle =\displaystyle 330\text{ cm}^3Evaluate
\displaystyle V\displaystyle =\displaystyle 1170-330Subtract the 2 volumes
\displaystyle =\displaystyle 840 \text{ cm}^3Evaluate
Idea summary

Some composite solids are made by joining simple solids together. We can find the volume of these composite solids by finding the volume of each of the pieces and adding them.

In the case where the composite solid is made by subtracting one simple solid from another, we can subtract volumes to find the volume of the composite solid.

Surface area of composite solids

The surface area of a composite solid is the sum of the areas of its faces.

In the case where the composite solid is also a prism, we can find its surface in the same way that we'd find the surface area of a prism.

If the composite solid is not a prism, we can find the surface area by considering the areas of each face of the solid and adding them up.

Examples

Example 3

Find the surface area of the house below.

A house with a width of 5 metres, length of 9 metres, height of 6 metres and a roof. Ask your teacher for more information.
Worked Solution
Create a strategy

Add all the faces together. The diagram below shows which faces are equal.

The faces of the house-shaped composite solid. Ask your teacher for more information.

Use Pythagoras' theorem, c^2=a^2+b^2, to find the width of the roof panels.

Apply the idea
\displaystyle \text{Front and back}\displaystyle =\displaystyle \left(\text{rectangle}+\text{triangle}\right)\times 2Add the areas and multiply by 2
\displaystyle =\displaystyle \left(6\times 5+\dfrac{1}{2}\times 5 \times 3\right)\times 2Substitute the dimensions
\displaystyle =\displaystyle 75\text{ m}^2Evaluate
\displaystyle \text{Side walls}\displaystyle =\displaystyle 9\times 6 \times 2Multiply the length and width and double
\displaystyle =\displaystyle 108\text{ m}^2Evaluate
\displaystyle \text{Base}\displaystyle =\displaystyle 5\times 9Multiply the length and width
\displaystyle =\displaystyle 45\text{ m}^2Evaluate

Using Pythagoras' theorem, we can find the width of the roof panels. On the given diagram, we can see two right-angled triangles where the shorter sides are a=3 and b=\dfrac{5}{2}=2.5.

\displaystyle c^2\displaystyle =\displaystyle 3^2+(2.5)^2Use Pythagoras' theorem
\displaystyle c^2\displaystyle =\displaystyle 15.25Evaluate
\displaystyle c\displaystyle =\displaystyle \sqrt{15.25}Square root both sides
\displaystyle \text{Roof sides}\displaystyle =\displaystyle 9\times \sqrt{15.25} \times 2Find the area of the 2 roof sides
\displaystyle =\displaystyle 18\sqrt{15.25}\text{ m}^2Evaluate
\displaystyle SA\displaystyle =\displaystyle 75+108+45+18\sqrt{15.25}Add all the areas
\displaystyle =\displaystyle 298.29\text{ m}^2Evaluate
Idea summary

The surface area of a composite solid is the sum of the areas of its faces.

Outcomes

MA5.2-11MG

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

MA5.2-12MG

applies formulas to calculate the volumes of composite solids composed of right prisms and cylinders

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