Nets and surface area of right prisms | Year 9

01 / LEARN

The idea, step by step

Unfold the outside

Surface area is the total area of the outside faces. A net lays those faces flat so you can count them without missing a back or bottom face. For a rectangular prism, opposite faces have equal areas. Write each face area before adding. All dimensions must use the same length unit.

A = 2 l w + 2 l h + 2 w h

Use the cross-section

A right prism has two congruent end faces and rectangular side faces. Add the perimeter of the cross-section, multiply it by the prism length, then add the two end areas. If a box is open, remove only the face that is missing; do not remove its volume. A drawing may look different after rotating, but the face areas do not change.

A = 2 B + P L

, where

B

is end area,

P

is end perimeter and

L

is prism length.

A common trap

Multiplying all three dimensions and calling the answer surface area.

Instead: Multiplying length, width and height gives volume. Find each face area and add for surface area.

02 / WORKED EXAMPLES

See the thinking

Example 1

Count pairs of faces

Find the surface area of the 6 cm by 4 cm by 3 cm prism.

The face pairs have areas $2 (6 \times 4) = 48$ , $2 (6 \times 3) = 36$ and $2 (4 \times 3) = 24$ square centimetres.
Add the three pairs: $48 + 36 + 24 = 108$ .

108 cm²

Example 2

A triangular cross-section

A right triangular prism has cross-section sides 3 cm, 4 cm and 5 cm. The 3 cm and 4 cm sides are perpendicular. Its length is 10 cm. Find its surface area.

One triangular end has area $B = \frac{1}{2} (3) (4) = 6$ cm².
The perimeter is $P = 3 + 4 + 5 = 12$ cm, so the side area is $12 \times 10 = 120$ cm².
Total area is $2 (6) + 120 = 132$ cm².

132 cm²

Check your understanding

How many faces must be counted in a closed rectangular prism?

Show answer

Six faces.

There are three pairs of opposite rectangular faces.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the surface area of a closed rectangular prism 4 cm long, 3 cm wide and 2 cm high.
2 marks
A cube has side length 5 cm. Find its total surface area.
2 marks
A right triangular prism has perpendicular cross-section sides 6 cm and 8 cm, a third side 10 cm, and length 7 cm. Find its surface area.
3 marks
An open-top rectangular box is 8 cm long, 5 cm wide and 3 cm high. Find the material area, ignoring thickness.
3 marks
A closed rectangular prism has length 5 cm, width 4 cm and surface area 94 cm². Find its height.
3 marks
A right prism has cross-section area 24 cm², cross-section perimeter 20 cm and length 9 cm. Find its surface area.
2 marks

Read the worked solutions

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

Find the surface area of a closed rectangular prism 4 cm long, 3 cm wide and 2 cm high.
2 marks
$2 (4 \times 3 + 4 \times 2 + 3 \times 2) = 2 (12 + 8 + 6) = 52$ cm².
52 cm²
A cube has side length 5 cm. Find its total surface area.
2 marks
Each square has area $25$ cm². Six faces give $6 \times 25 = 150$ cm².
150 cm²
A right triangular prism has perpendicular cross-section sides 6 cm and 8 cm, a third side 10 cm, and length 7 cm. Find its surface area.
3 marks
Two ends: $2 (\frac{1}{2} \times 6 \times 8) = 48$ cm².
Sides: $(6 + 8 + 10) \times 7 = 168$ cm². Total $216$ cm².
216 cm²
An open-top rectangular box is 8 cm long, 5 cm wide and 3 cm high. Find the material area, ignoring thickness.
3 marks
Base $8 \times 5 = 40$ ; two long walls $2 (8 \times 3) = 48$ ; two short walls $2 (5 \times 3) = 30$ .
Total $40 + 48 + 30 = 118$ cm².
118 cm²
A closed rectangular prism has length 5 cm, width 4 cm and surface area 94 cm². Find its height.
3 marks
$2 (20 + 5 h + 4 h) = 94$ . Thus $40 + 18 h = 94$ , so $18 h = 54$ and $h = 3$ cm.
3 cm
A right prism has cross-section area 24 cm², cross-section perimeter 20 cm and length 9 cm. Find its surface area.
2 marks
$A = 2 B + P L = 2 (24) + 20 (9) = 48 + 180 = 228$ cm².
228 cm²

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Year 9 surface area and volume of right prisms and cylinders, capacity, absolute/relative/percentage measurement error, precision and practical measurement problems.

Official curriculum source ↗

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