Prism volume and capacity | Year 8

01 / LEARN

Understand the idea.

Build volume from area

A right prism repeats one cross-section along its length, giving volume $V = A L$ . First find the end's area $A$ , then multiply by the perpendicular distance $L$ between the ends. For a right cylinder this becomes $V = π r^{2} h$ .

Use consistent length units throughout. An area in ${cm}^{2}$ multiplied by a length in $cm$ produces ${cm}^{3}$ . A circumference is only a length, so multiplying circumference by height cannot give a volume.

V = A L

Connect volume and capacity

Capacity is the volume a container can hold. The connections are $1 {cm}^{3} = 1 mL$ , $1000 {cm}^{3} = 1 L$ and $1 m^{3} = 1000 L$ . Calculate the internal volume before converting its unit.

Check whether dimensions describe the inside or outside of a container. Thick walls occupy space, so outside measurements can overstate capacity. Use supplied internal dimensions, and keep an exact multiple of $π$ when an exact capacity is requested.

Volume and capacity
Volume	Capacity
1 cm³	1 mL
1000 cm³	1 L
1 m³	1000 L

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A triangular prism

A triangular prism has cross-section base 6 cm, perpendicular height 4 cm and length 9 cm. Find its volume.

Worked solution

1
Find the triangular cross-sectional area.
$A = \frac{1}{2} \times 6 \times 4 = 12 {cm}^{2}$
The base and its perpendicular height describe the repeated triangular end.
2
Use the distance between the ends.
$V = A L = 12 \times 9$
The same cross-section continues for the full 9-centimetre prism length.
3
Calculate the volume.
$V = 108 {cm}^{3}$
Square centimetres multiplied by centimetres give cubic centimetres.

Answer

108 cm³.

EXAMPLE 02

A cylinder

A cylindrical container has internal radius 3 cm and height 7 cm. Find its volume exactly and its capacity in mL.

Worked solution

1
Find the circular cross-sectional area.
$A = π {(3)}^{2} = 9 π {cm}^{2}$
The internal radius determines the area available inside each layer.
2
Multiply by the internal height.
$V = 9 π \times 7 = 63 π {cm}^{3}$
The cylinder repeats that circular area over a height of 7 centimetres.
3
Convert volume to capacity.
$Capacity = 63 π mL$
Each cubic centimetre holds one millilitre, so the numerical value stays the same.

Answer

$63 π$ cm³, equivalent to $63 π$ mL.

03 / YOUR TURN

Try it yourself.

Check your understanding

A prism has cross-sectional area 12 cm² and length 5 cm. What is its volume?

Show answer & reasoning

60 cm³.

Multiply the repeated cross-sectional area by the prism length: $V = 12 \times 5 = 60 {cm}^{3}$ . The units are cubic because ${cm}^{2} \times cm = {cm}^{3}$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Find the volume of a rectangular prism 8 cm by 3 cm by 2 cm.
2 marks
A prism has cross-sectional area 15 m² and length 4 m. Find its volume.
2 marks
A triangular prism has cross-section base 10 cm, perpendicular height 6 cm and length 8 cm. Find its volume.
3 marks
Find the volume of a cylinder with radius 2 m and height 5 m.
2 marks
A container holds 3500 cm³. Express its capacity in litres.
2 marks
A rectangular tank holds 240 L. Its internal base is 0.8 m by 0.5 m. Find its internal height.
4 marks

Read the worked solutions

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

Find the volume of a rectangular prism 8 cm by 3 cm by 2 cm.
2 marks
$V = 8 (3) (2) = 48$ cm³.
48 cm³.
A prism has cross-sectional area 15 m² and length 4 m. Find its volume.
2 marks
$V = 15 (4) = 60$ m³.
60 m³.
A triangular prism has cross-section base 10 cm, perpendicular height 6 cm and length 8 cm. Find its volume.
3 marks
Cross-sectional area is $\frac{1}{2} (10) (6) = 30$ cm².
Volume is $30 (8) = 240$ cm³.
240 cm³.
Find the volume of a cylinder with radius 2 m and height 5 m.
2 marks
$V = π {(2)}^{2} (5) = 20 π$ m³.
$20 π$ m³.
A container holds 3500 cm³. Express its capacity in litres.
2 marks
$3500 \div 1000 = 3.5$ L.
3.5 L.
A rectangular tank holds 240 L. Its internal base is 0.8 m by 0.5 m. Find its internal height.
4 marks
240 L is 0.24 m³.
Base area is $0.8 (0.5) = 0.4$ m².
Height is $\frac{0.24}{0.4} = 0.6$ m.
0.6 m.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Circumference and area of circles; perimeter/area of composite shapes; choosing and converting area units; cross-sections of prisms; prism volume and capacity; mixed measurement applications.

Official curriculum source ↗

In this lesson

Understand the idea.

Build volume from area

Connect volume and capacity

See the method in action.

A triangular prism

Find the triangular cross-sectional area.

Use the distance between the ends.

Calculate the volume.

A cylinder

Find the circular cross-sectional area.

Multiply by the internal height.

Convert volume to capacity.

Try it yourself.

Check your understanding

Keep practising