Volume of composite solids | Year 10

01 / LEARN

The idea, step by step

Add or subtract space

Split the object into pieces that do not overlap. Calculate each volume and add. If material has been removed, subtract the cavity volume from the original solid.

Unlike surface area, volume does not lose a contact face when solids join. A face has no thickness, so it occupies no volume.

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 the radii share an axis and the height must be common to both cylinders. Use radii rather than diameters in the formula.

Use internal dimensions for capacity. One cubic centimetre equals one millilitre, one thousand cubic centimetres equals one litre, and one cubic metre equals one thousand litres.

V = π (R^{2} - r^{2}) h

A common trap

Subtracting a shared face from a volume calculation.

Instead: Subtract actual removed or overlapping space, not a zero-thickness face.

02 / WORKED EXAMPLES

See the thinking

Example 1

Stepped block

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

Volumes:8×5×3=120 and 4×5×2=40 cm³.
Add 120+40=160 cm³.

160 cm³

Example 2

Tank with a dome

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

Cylinder $= 72 π$ .
Hemisphere $= \frac{2}{3} π (3^{3}) = 18 π$ .
Total $= 90 π {cm}^{3}$ .

90 π {cm}^{3}

Check your understanding

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

Show answer

1.6 L

Its volume is 1600 cm³, or 1.6 L.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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
20×10×8=1600 cm³.
1600÷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 ↗

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