Rectangular/triangular prism volume | Year 7

01 / LEARN

Understand the idea.

Volume counts space inside

Volume measures three-dimensional space in cubic units. A prism keeps the same cross-section along its length. Imagine stacking identical thin layers: the cross-section area measures one layer, and the perpendicular distance between the ends measures how far the layers extend.

For a right rectangular prism, find a rectangular end area and multiply by the prism length. This is equivalent to multiplying its length, width and height. Check that all three dimensions use the same unit before multiplying.

V = A_{cross-section} \times l

Triangular prisms use triangle area first

For a triangular prism, calculate the triangular end area first using $\frac{1}{2} b h$ . Then multiply by the prism length. The triangle’s perpendicular height lies in the end face; the prism length runs between the two ends. They play different roles.

Area units multiplied by length units give cubic units: ${cm}^{2} \times cm = {cm}^{3}$ . For capacity, $1 {cm}^{3}$ corresponds to $1 mL$ , but use internal dimensions when calculating how much a container can actually hold.

V = \frac{1}{2} b h \times l

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Rectangular prism

Find the volume of a cuboid 8 cm long, 3 cm deep and 4 cm high.

Worked solution

1
Find a rectangular end area.
$A = 3 \times 4 = 12 {cm}^{2}$
Depth and height describe the end perpendicular to the eight-centimetre length.
2
Extend that cross-section along the prism.
$V = A \times l = 12 \times 8$
Each layer has the same cross-section area.
3
State the volume in cubic units.
$V = 96 {cm}^{3}$
Square centimetres multiplied by centimetres give cubic centimetres.

Answer

$96 {cm}^{3}$

EXAMPLE 02

Triangular prism

A triangular prism has end base 6 cm, perpendicular end height 4 cm and length 10 cm. Find its volume.

Worked solution

1
Find the triangular cross-section area.
$A = \frac{1}{2} \times 6 \times 4 = 12 {cm}^{2}$
The six-centimetre base and four-centimetre perpendicular height describe the triangular end.
2
Multiply by the length between the ends.
$V = 12 \times 10$
The triangular cross-section continues unchanged for ten centimetres.
3
Write the volume.
$V = 120 {cm}^{3}$
The result measures space rather than the area of a face.

Answer

$120 {cm}^{3}$

03 / YOUR TURN

Try it yourself.

Check your understanding

A prism has cross-section area 15 cm² and length 8 cm. Find its volume.

Show answer & reasoning

120 cm³

Multiply the cross-section area by the prism length: $15 \times 8 = 120$ . Since ${cm}^{2} \times cm = {cm}^{3}$ , the volume is $120 {cm}^{3}$ .

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6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the volume of a cuboid 5 cm by 4 cm by 3 cm.
2 marks
A prism has cross-section area 18 m² and length 7 m. Find its volume.
1 mark
Find the volume of a triangular prism with end base 8 cm, perpendicular end height 3 cm and prism length 12 cm.
3 marks
A cuboid has volume 240 cm³. Its base is 8 cm by 5 cm. Find its height.
3 marks
The inside of a rectangular container is 10 cm by 8 cm by 5 cm. How many mL can it hold?
2 marks
Two prisms have the same cross-section area. One is 6 cm long and the other is 15 cm long. The shorter has volume 48 cm³. Find the longer volume.
3 marks

Read the worked solutions

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

Find the volume of a cuboid 5 cm by 4 cm by 3 cm.
2 marks
$5 \times 4 \times 3 = 60$ cm³.
60 cm³
A prism has cross-section area 18 m² and length 7 m. Find its volume.
1 mark
$18 \times 7 = 126$ m³.
126 m³
Find the volume of a triangular prism with end base 8 cm, perpendicular end height 3 cm and prism length 12 cm.
3 marks
$End area = \frac{1}{2} \times 8 \times 3 = 12$ cm².
$Volume = 12 \times 12 = 144$ cm³.
144 cm³
A cuboid has volume 240 cm³. Its base is 8 cm by 5 cm. Find its height.
3 marks
$Base area = 8 \times 5 = 40$ cm².
$Height = volume \div base area$ .
$240 \div 40 = 6$ cm.
6 cm
The inside of a rectangular container is 10 cm by 8 cm by 5 cm. How many mL can it hold?
2 marks
$Volume = 10 \times 8 \times 5 = 400$ cm³.
$1 {cm}^{3} = 1 mL$ .
400 mL
Two prisms have the same cross-section area. One is 6 cm long and the other is 15 cm long. The shorter has volume 48 cm³. Find the longer volume.
3 marks
$Cross-section area = 48 \div 6 = 8$ cm².
$Longer volume = 8 \times 15 = 120$ cm³.
120 cm³

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 7 measurement and spatial representation: areas of triangles/parallelograms, prism volumes, circle circumference, nets and views. Precision is integrated into practical measurement; circle area is not introduced here.

Official curriculum source ↗

In this lesson

Understand the idea.

Volume counts space inside

Triangular prisms use triangle area first

See the method in action.

Rectangular prism

Find a rectangular end area.

Extend that cross-section along the prism.

State the volume in cubic units.

Triangular prism

Find the triangular cross-section area.

Multiply by the length between the ends.

Write the volume.

Try it yourself.

Check your understanding

Keep practising