Cylinder surface area | Year 9

01 / LEARN

The idea, step by step

Unroll the curved face

The curved wall of a cylinder unrolls to a rectangle. Its length is the circle circumference, not its diameter. Its width is the cylinder height. The curved area is therefore circumference multiplied by height. A closed cylinder also has two circular ends.

A_{c u r v e d} = 2 π r h, A_{c l o s e d} = 2 π r^{2} + 2 π r h

Match the formula to the object

A tin can is usually closed at both ends; an open tank has one circular base. A label wrapped around a can covers only the curved surface. Sketch and list the surfaces before choosing a formula. If a diameter is supplied, halve it to get the radius. Keep pi in the calculation until the final rounding.

A common trap

Using diameter in place of radius, or automatically counting two ends.

Instead: Identify the radius and the surfaces that actually exist. Each circular end contributes one circle area.

02 / WORKED EXAMPLES

See the thinking

Example 1

A closed cylinder

Find the total surface area of a closed cylinder with radius 3 cm and height 8 cm.

Two ends have area $2 π (3^{2}) = 18 π$ cm².
The curved face has area $2 π (3) (8) = 48 π$ cm².
Total area is $66 π$ cm², approximately $207.3$ cm².

66 π

cm², or 207.3 cm² to one decimal place.

Example 2

One end is missing

A cylindrical tank has radius 2 m and height 5 m. It has a base but no lid. Find the area of material needed.

Include one base: $π (2^{2}) = 4 π$ m².
Include the wall: $2 π (2) (5) = 20 π$ m².
Total $4 π + 20 π = 24 π$ m².

24 π

m²

Check your understanding

What is the unrolled length of the curved face when the radius is 4 cm?

Show answer

$8 π$ cm.

The rectangle length is the circumference $2 π r = 2 π (4)$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the total surface area of a closed cylinder with radius 4 cm and height 6 cm. Give an exact answer.
2 marks
A closed cylinder has diameter 10 cm and height 7 cm. Find its surface area.
2 marks
Find the curved surface area of a cylinder with radius 3 cm and height 12 cm.
2 marks
An open cylindrical container has radius 5 cm and height 10 cm. It has one base. Find its material area.
3 marks
A label covers only the wall of a can with radius 2 cm and height 6 cm. Find the label area.
2 marks
A closed cylinder has radius 3 cm and total surface area $90 π$ cm². Find its height.
3 marks

Read the worked solutions

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

Find the total surface area of a closed cylinder with radius 4 cm and height 6 cm. Give an exact answer.
2 marks
$2 π (4^{2}) + 2 π (4) (6) = 32 π + 48 π = 80 π$ cm².
$80 π$ cm²
A closed cylinder has diameter 10 cm and height 7 cm. Find its surface area.
2 marks
The radius is 5 cm. Area $= 2 π (25) + 2 π (5) (7) = 50 π + 70 π = 120 π$ cm².
$120 π$ cm²
Find the curved surface area of a cylinder with radius 3 cm and height 12 cm.
2 marks
$2 π r h = 2 π (3) (12) = 72 π$ cm².
$72 π$ cm²
An open cylindrical container has radius 5 cm and height 10 cm. It has one base. Find its material area.
3 marks
One base $25 π$ plus wall $2 π (5) (10) = 100 π$ gives $125 π$ cm².
$125 π$ cm²
A label covers only the wall of a can with radius 2 cm and height 6 cm. Find the label area.
2 marks
Only curved area is needed: $2 π (2) (6) = 24 π$ cm².
$24 π$ cm²
A closed cylinder has radius 3 cm and total surface area $90 π$ cm². Find its height.
3 marks
$18 π + 6 π h = 90 π$ . Divide by pi: $18 + 6 h = 90$ , so $h = 12$ cm.
12 cm

Checking saved progress…

Continue learning

Prism/cylinder volume and capacity

Next lesson

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 ↗

Back to Surface area, volume and measurement