Surface Area of a Cylinder Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Surface Area of a Cylinder.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The total area of the surface of a cylinder, consisting of two circular bases and a rectangular lateral surface that wraps around.

Imagine peeling the label off a can of soup. The label is a rectangle whose width is the circumference of the can ( $2\pi r$ ) and whose height is the can's height ( $h$ ). Add the two circular lids (top and bottom), and you have the total surface area.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Surface area of a cylinder is two circular bases plus the rectangle that wraps around the side.

Common stuck point: The procedure for surface area of a cylinder is the easy part; the trap is dropping the lateral term. Asking "Am I covering both circular ends and the curved side of a cylinder?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I covering both circular ends and the curved side of a cylinder?

Worked Examples

Example 1

easy

A cylinder has radius 5 cm and height 8 cm. Find its total surface area. Leave your answer in terms of

\pi

Cylinder with r = 5 cm, h = 8 cm

Answer

SA = 130\pi

cm².

First step

Step 1: Write the formula:

SA = 2\pi r^2 + 2\pi rh

Full solution

2
Step 2: The two circular bases contribute: $2\pi r^2 = 2\pi(5)^2 = 50\pi$ cm².
3
Step 3: The lateral (curved) surface contributes: $2\pi rh = 2\pi(5)(8) = 80\pi$ cm².
4
Step 4: Total: $SA = 50\pi + 80\pi = 130\pi$ cm².

The cylinder's surface area has two parts: the two circular ends (

2\pi r^2

) and the curved lateral surface (

2\pi rh

). The lateral surface, when unrolled, forms a rectangle of width

2\pi r

(the circumference) and height

h

Example 2

medium

A cylindrical can has a total surface area of

150\pi

cm² and a radius of 5 cm. Find the height.

Find the height h given SA = 150π cm²

Example 3

medium

Find the total surface area of a cylinder with radius 4 cm and height 10 cm.

Cylinder with r = 4 cm, h = 10 cm

Example 4

easy

A can has diameter

8

cm and height

10

cm. Find its total surface area in terms of

\pi

Can: diameter 8 cm → r = 4 cm, h = 10 cm

Example 5

medium

A cylindrical water tank (closed both ends) has

r = 10

ft,

h = 14

ft. How much paint, in square feet, covers the outside? (Use

\pi \approx 3.14

Example 6

medium

A cylindrical silo has

r = 4

m and

h = 10

m. Paint covers

50

^2

per gallon. How many gallons cover its lateral surface? (Use

\pi \approx 3.14

Example 7

hard

A cylindrical tube (open both ends) has outer radius

6

, inner radius

5

, length

12

. Find the total surface area in terms of

\pi

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the lateral surface area only (not the bases) of a cylinder with radius 3 cm and height 7 cm. Leave your answer in terms of

\pi

Find the lateral surface area (no bases)

Example 2

hard

A cylinder's height equals its diameter. If the total surface area is

108\pi

cm², find the radius.

Example 3

easy

Find the total surface area of a cylinder with radius

1

and height

5

, in terms of

\pi

Example 4

easy

Find the lateral surface area of a cylinder with radius

5

and height

8

, in terms of

\pi

Example 5

easy

A cylindrical mug (open top, closed bottom) has radius

3

and height

8

. Find its outer surface area in terms of

\pi

Example 6

easy

Find the total surface area of a cylinder with

r = 7

and

h = 3

, in terms of

\pi

Cylinder: r

Example 7

medium

A cylinder has lateral area

30\pi

and radius

3

. Find its height.

Find height h given r

Example 8

medium

Two cylinders have the same radius

r = 4

but heights

5

and

9

. Find the ratio of their lateral surface areas.

Example 9

medium

If a cylinder's radius doubles but height stays the same, by what factor does the total surface area change?

Example 10

medium

A cylindrical label (no overlap) wraps a can of radius

4

and height

11

. Find the area of the label in terms of

\pi

Example 11

medium

A pipe (hollow open cylinder) has radius

2

and length

15

. Find its surface area in terms of

\pi

(lateral only).

Example 12

medium

Express the total surface area of a cylinder of fixed volume

V

and radius

r

in terms of

r

Example 13

hard

A cylinder's height is twice its radius. If its total surface area is

54\pi

, find

r

Example 14

hard

A cylinder of radius

5

and height

12

is melted and reformed into a cube. Find the cube's surface area to the nearest whole unit (use

\pi \approx 3.14

Example 15

hard

A cylinder's total surface area equals its volume numerically. If

r = 3

, find

h

Example 16

hard

Two open cylindrical cups have the same lateral surface area. The first has

r = 3

h = 8

. If the second has

r = 4

, find its height.

Example 17

hard

A cylinder has total surface area

24\pi

. If

r = 1

, find

h

Example 18

challenge

A closed cylindrical can holds volume

250\pi

. Find the radius minimizing its total surface area.

Example 19

challenge

A closed cylindrical can of fixed surface area

S

has volume

V(r) = \frac{r(S - 2\pi r^2)}{2}

. Find the value of

r

that maximizes

V

← Back to Surface Area of a Cylinder Formula Explained Practice Mode

Related Concepts

Area of a Circle Surface Area Circumference Surface Area of a Cylinder Geometric Modeling

Background Knowledge

These ideas may be useful before you work through the harder examples.

area of circlesurface areacircumference