Nets and surface area of right prisms | Year 9

01 / LEARN

Understand the idea.

Unfold the outside

Surface area measures the outside covering of a solid. Imagine cutting along edges and flattening the faces into a net: every exposed face must appear exactly once, including faces hidden in the drawing.

A rectangular prism has three pairs of equal rectangular faces. Calculate each pair using matching length units, then add. Multiplying two lengths produces square units, so an answer for surface area must use units such as ${cm}^{2}$ .

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

Use the cross-section

For a right prism, the two ends share area $B$ . Each side rectangle has the prism length $L$ and one edge of the end shape. Adding those rectangles gives the end perimeter multiplied by the length: $P L$ .

The closed prism therefore has area $2 B + P L$ . For an open container, first identify the missing face and omit its area. Rotating the object changes which faces you see, but does not change the total covering.

A = 2 B + P L

, where

B

is end area,

P

is end perimeter and

L

is prism length.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Count pairs of faces

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

Worked solution

1
Find the top and bottom areas.
$2 (6 \times 4) = 48 {cm}^{2}$
These two equal faces each measure length by width.
2
Find the front and back areas.
$2 (6 \times 3) = 36 {cm}^{2}$
These two equal faces each measure length by height.
3
Find the remaining two faces.
$2 (4 \times 3) = 24 {cm}^{2}$
These equal side faces measure width by height.
4
Add all six faces.
$A = 48 + 36 + 24 = 108 {cm}^{2}$
The three pairs account for every face of the closed prism.

Answer

108 cm²

EXAMPLE 02

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.

Worked solution

1
Calculate one triangular end.
$B = \frac{1}{2} \times 3 \times 4 = 6 {cm}^{2}$
The perpendicular sides provide the triangle base and height.
2
Find the end perimeter.
$P = 3 + 4 + 5 = 12 cm$
All three triangle edges generate rectangular side faces.
3
Calculate the side area.
$P L = 12 \times 10 = 120 {cm}^{2}$
Each side rectangle extends the full prism length.
4
Include both ends.
$A = 2 (6) + 120 = 132 {cm}^{2}$
A closed triangular prism has two congruent ends as well as its three side faces.

Answer

132 cm²

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Six faces.

A closed rectangular prism has three pairs of opposite faces: top and bottom, front and back, and the two sides. That makes $3 \times 2 = 6$ faces.

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Unfold the outside

Use the cross-section

See the method in action.

Count pairs of faces

Find the top and bottom areas.

Find the front and back areas.

Find the remaining two faces.

Add all six faces.

A triangular cross-section

Calculate one triangular end.

Find the end perimeter.

Calculate the side area.

Include both ends.

Try it yourself.

Check your understanding

Keep practising