Volume of composite solids | Year 10

01 / LEARN

Understand the idea.

Add or subtract space

For composite volume, split the occupied space into non-overlapping solids and add their volumes. If a cavity is removed from a solid, subtract the cavity's volume instead. Sketch the pieces before selecting formulas.

A shared contact face has no thickness, so it occupies no volume. Joining solids along faces does not require the double-face subtraction used for surface area; only overlapping or removed three-dimensional space must be subtracted.

V_{c o m p o s i t e} = V_{a d d e d} - V_{r e m o v e d}

Check the measurements

For a hollow cylinder, subtract the inner cylinder from the outer one using their common height and axis. Check whether the given lengths are radii or diameters before squaring them in the volume formula.

Capacity uses the space available inside a container, not its exterior dimensions. Convert after calculating volume: $1 {cm}^{3} = 1 mL$ , $1000 {cm}^{3} = 1 L$ and $1 m^{3} = 1000 L$ .

V = π (R^{2} - r^{2}) h

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Stepped block

Two non-overlapping cuboids measure 8 by 5 by 3 cm and 4 by 5 by 2 cm. Find combined volume.

Worked solution

1
Calculate the larger cuboid volume.
$V_{1} = 8 \cdot 5 \cdot 3 = 120 {cm}^{3}$
A cuboid occupies length times width times height.
2
Calculate the second cuboid volume.
$V_{2} = 4 \cdot 5 \cdot 2 = 40 {cm}^{3}$
The second component uses its own three dimensions.
3
Combine the non-overlapping pieces.
$V = 120 + 40 = 160 {cm}^{3}$
The question states that their occupied spaces do not overlap.

Answer

160 cm³

EXAMPLE 02

Tank with a dome

A radius 3 cm, height 8 cm cylinder has a hemisphere of radius 3 cm attached. Find total volume.

Worked solution

1
Calculate the cylinder volume.
$V_{cylinder} = π {(3)}^{2} (8) = 72 π {cm}^{3}$
The circular base area is multiplied by the cylinder height.
2
Calculate half the sphere volume.
$V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3} π {(3)}^{3} = 18 π {cm}^{3}$
The attached dome is a hemisphere with the same radius.
3
Add the component volumes.
$V = 72 π + 18 π = 90 π {cm}^{3}$
The matching circular face has zero thickness and the components do not overlap.

Answer

$90 π {cm}^{3}$

03 / YOUR TURN

Try it yourself.

Check your understanding

A box measures 20 by 10 by 8 cm internally. Find its capacity.

Show answer & reasoning

1.6 L

The internal volume is $20 \cdot 10 \cdot 8 = 1600 {cm}^{3}$ . Dividing by $1000$ gives a capacity of $1.6 L$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

A 12 by 6 by 4 cm block has a 4 by 2 by 4 cm hole removed. Find remaining volume.
2 marks
Combine non-overlapping blocks 10 by 8 by 3 cm and 3 by 4 by 5 cm. Find volume.
2 marks
A hollow cylinder has outer radius 5 cm, inner radius 3 cm and height 12 cm. Find material volume.
2 marks
A cylinder radius 3 cm, height 8 cm supports a matching cone of height 4 cm. Find total volume.
3 marks
A semicylindrical solid has radius 2 cm and length 10 cm. Find volume.
2 marks
A container measures 20 by 10 by 8 cm internally. Find capacity in litres.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A 12 by 6 by 4 cm block has a 4 by 2 by 4 cm hole removed. Find remaining volume.
2 marks
$288 - 32 = 256$ .
256 cm³
Combine non-overlapping blocks 10 by 8 by 3 cm and 3 by 4 by 5 cm. Find volume.
2 marks
$240 + 60 = 300$ .
300 cm³
A hollow cylinder has outer radius 5 cm, inner radius 3 cm and height 12 cm. Find material volume.
2 marks
$π (25 - 9) (12) = 192 π$ .
$192 π {cm}^{3}$
A cylinder radius 3 cm, height 8 cm supports a matching cone of height 4 cm. Find total volume.
3 marks
Cylinder $72 π$ ; cone $12 π$ .
Total $84 π$ .
$84 π {cm}^{3}$
A semicylindrical solid has radius 2 cm and length 10 cm. Find volume.
2 marks
$\frac{1}{2} π (2^{2}) (10) = 20 π$ .
$20 π {cm}^{3}$
A container measures 20 by 10 by 8 cm internally. Find capacity in litres.
2 marks
$20 \times 10 \times 8 = 1600$ cm³.
$1600 \div 1000 = 1.6$ L.
1.6 L

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Surface area of composite solids; volume of composite solids; proportion/scaling of objects; related length/area/volume factors; measurement error; the effect of rounding and repeated approximations; practical design models; evaluating assumptions and suitable precision.

Official curriculum source ↗

In this lesson

Understand the idea.

Add or subtract space

Check the measurements

See the method in action.

Stepped block

Calculate the larger cuboid volume.

Calculate the second cuboid volume.

Combine the non-overlapping pieces.

Tank with a dome

Calculate the cylinder volume.

Calculate half the sphere volume.

Add the component volumes.

Try it yourself.

Check your understanding

Keep practising