Surface area | Year 11 | Ascent Library

01 / LEARN

Understand the idea.

Understand the idea

Surface area is the total area of the faces the question asks to cover. A closed cuboid has six faces arranged in three equal pairs. Sketching its net makes it easier to identify every face without counting any twice.

For a cylinder, imagine cutting the curved wall vertically and unrolling it. The resulting rectangle has length equal to the circumference, $2 π r$ , and width equal to the height, $h$ . Each circular end is a separate area of $π r^{2}$ .

Choose and apply a method

List the included surfaces before choosing a formula. A closed cylinder has a wall and two ends; an open-top tank has a wall and one base. The wording 'inside' also tells you which dimensions and faces matter.

Use consistent length units, then report square units. For joined solids, shared internal faces are not exposed surfaces. Keep expressions containing $π$ exact until the final approximation, so rounding does not change the total unnecessarily.

\begin{aligned} S A_{cuboid} & = 2 (l w + l h + w h) \\ S A_{closed cylinder} & = 2 π r^{2} + 2 π r h \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Cover a cuboid

Find surface area of a closed cuboid 6 cm by 4 cm by 3 cm.

Worked solution

1
Calculate the three distinct face areas.
$\begin{aligned} l w & = 6 \times 4 = 24 \\ l h & = 6 \times 3 = 18 \\ w h & = 4 \times 3 = 12 \end{aligned} {(cm}^{2})$
Every cuboid face is one of these three rectangles.
2
Include both faces of each type.
$S A = 2 (24 + 18 + 12)$
The cuboid is closed, so all three matching pairs are included.
3
Evaluate the total.
$S A = 108 {cm}^{2}$
The areas add because the six faces do not overlap in a net.

Answer

108 cm $2$ .

EXAMPLE 02

Open cylinder

A cylindrical tank has radius 2 m and height 5 m. Find the inside base and wall area, excluding the open top.

Worked solution

1
Calculate the single base area.
$A_{base} = π {(2)}^{2} = 4 π m^{2}$
The open tank has one circular inside base and no top.
2
Calculate the curved wall area.
$A_{wall} = 2 π (2) (5) = 20 π m^{2}$
The unrolled wall has circumference times height as its area.
3
Add the included surfaces.
$S A = 4 π + 20 π = 24 π m^{2}$
Only the base and wall require coverage.
4
Approximate the exact area.
$S A \approx 75.40 m^{2}$
Round after adding the exact areas, rather than rounding each part first.

Answer

$24 π$ m $2$ , approximately 75.40 m $2$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the surface area of a cube of side 4 cm.

Show answer & reasoning

96 cm $2$ .

A cube has six equal square faces. Each has area $4^{2} = 16 {cm}^{2}$ , so the total surface area is $6 \times 16 = 96 {cm}^{2}$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find surface area of a cube with side 5 cm.
2 marks
Find surface area of a closed 8 cm by 3 cm by 2 cm cuboid.
2 marks
An open rectangular box has base 10 cm by 6 cm and height 4 cm. Find its material area.
2 marks
Find curved surface area of a cylinder with radius 3 cm and height 7 cm. Give an exact answer.
2 marks
Find total surface area of a closed cylinder with radius 3 cm and height 7 cm.
2 marks
Two cubes, each of side 2 cm, are joined face-to-face. Find exposed area.
2 marks

Read the worked solutions

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

Find surface area of a cube with side 5 cm.
2 marks
$6 \times 5^{2} = 150$ .
150 cm $2$ .
Find surface area of a closed 8 cm by 3 cm by 2 cm cuboid.
2 marks
$2 (24 + 16 + 6) = 92$ .
92 cm $2$ .
An open rectangular box has base 10 cm by 6 cm and height 4 cm. Find its material area.
2 marks
$60 + 2 (40) + 2 (24) = 188$ .
188 cm $2$ .
Find curved surface area of a cylinder with radius 3 cm and height 7 cm. Give an exact answer.
2 marks
$2 π (3) (7) = 42 π$ .
$42 π$ cm $2$ .
Find total surface area of a closed cylinder with radius 3 cm and height 7 cm.
2 marks
$2 π {(3)}^{2} + 2 π (3) (7) = 18 π + 42 π = 60 π$ .
$60 π$ cm $2$ .
Two cubes, each of side 2 cm, are joined face-to-face. Find exposed area.
2 marks
Separate area $= 2 \times 6 \times 2^{2} = 48$ .
Two hidden joining faces total $2 \times 2^{2} = 8$ .
Exposed area $= 48 - 8 = 40$ cm $2$ .
40 cm $2$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics measurement skills and applications; original explanatory lessons and papers.

Official curriculum source ↗

In this lesson

Understand the idea.

Understand the idea

Choose and apply a method

See the method in action.

Cover a cuboid

Calculate the three distinct face areas.

Include both faces of each type.

Evaluate the total.

Open cylinder

Calculate the single base area.

Calculate the curved wall area.

Add the included surfaces.

Approximate the exact area.

Try it yourself.

Check your understanding

Keep practising