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 (2πr2\pi r) and whose height is the can's height (hh). Add the two circular lids (top and bottom), and you have the total surface area.

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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 π\pi.

Answer

SA=130πSA = 130\pi cm².

First step

1
Step 1: Write the formula: SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi rh.

Full solution

  1. 2
    Step 2: The two circular bases contribute: 2πr2=2π(5)2=50π2\pi r^2 = 2\pi(5)^2 = 50\pi cm².
  2. 3
    Step 3: The lateral (curved) surface contributes: 2πrh=2π(5)(8)=80π2\pi rh = 2\pi(5)(8) = 80\pi cm².
  3. 4
    Step 4: Total: SA=50π+80π=130πSA = 50\pi + 80\pi = 130\pi cm².
The cylinder's surface area has two parts: the two circular ends (2πr22\pi r^2) and the curved lateral surface (2πrh2\pi rh). The lateral surface, when unrolled, forms a rectangle of width 2πr2\pi r (the circumference) and height hh.

Example 2

medium
A cylindrical can has a total surface area of 150π150\pi cm² and a radius of 5 cm. Find the height.

Example 3

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

Example 4

easy
A can has diameter 88 cm and height 1010 cm. Find its total surface area in terms of π\pi.

Example 5

medium
A cylindrical water tank (closed both ends) has r=10r = 10 ft, h=14h = 14 ft. How much paint, in square feet, covers the outside? (Use π≈3.14\pi \approx 3.14.)

Example 6

medium
A cylindrical silo has r=4r = 4 m and h=10h = 10 m. Paint covers 5050 m2^2 per gallon. How many gallons cover its lateral surface? (Use π≈3.14\pi \approx 3.14.)

Example 7

hard
A cylindrical tube (open both ends) has outer radius 66, inner radius 55, length 1212. Find the total surface area in terms of π\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 π\pi.

Example 2

hard
A cylinder's height equals its diameter. If the total surface area is 108π108\pi cm², find the radius.

Example 3

easy
Find the total surface area of a cylinder with radius 11 and height 55, in terms of π\pi.

Example 4

easy
Find the lateral surface area of a cylinder with radius 55 and height 88, in terms of π\pi.

Example 5

easy
A cylindrical mug (open top, closed bottom) has radius 33 and height 88. Find its outer surface area in terms of π\pi.

Example 6

easy
Find the total surface area of a cylinder with r=7r = 7 and h=3h = 3, in terms of π\pi.

Example 7

medium
A cylinder has lateral area 30π30\pi and radius 33. Find its height.

Example 8

medium
Two cylinders have the same radius r=4r = 4 but heights 55 and 99. 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 44 and height 1111. Find the area of the label in terms of π\pi.

Example 11

medium
A pipe (hollow open cylinder) has radius 22 and length 1515. Find its surface area in terms of π\pi (lateral only).

Example 12

medium
Express the total surface area of a cylinder of fixed volume VV and radius rr in terms of rr.

Example 13

hard
A cylinder's height is twice its radius. If its total surface area is 54π54\pi, find rr.

Example 14

hard
A cylinder of radius 55 and height 1212 is melted and reformed into a cube. Find the cube's surface area to the nearest whole unit (use π≈3.14\pi \approx 3.14).

Example 15

hard
A cylinder's total surface area equals its volume numerically. If r=3r = 3, find hh.

Example 16

hard
Two open cylindrical cups have the same lateral surface area. The first has r=3r = 3, h=8h = 8. If the second has r=4r = 4, find its height.

Example 17

hard
A cylinder has total surface area 24π24\pi. If r=1r = 1, find hh.

Example 18

challenge
A closed cylindrical can holds volume 250π250\pi. Find the radius minimizing its total surface area.

Example 19

challenge
A closed cylindrical can of fixed surface area SS has volume V(r)=r(S−2πr2)2V(r) = \frac{r(S - 2\pi r^2)}{2}. Find the value of rr that maximizes VV.
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Background Knowledge

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

area of circlesurface areacircumference