Surface Area of a Cylinder

16 min

Let's begin

Niyonzima's family catches rainwater from the roof in a round metal tank. The tank is starting to rust, so they plan to paint it. A tin of paint says it covers 5 m². How many tins will they need?

To answer, they first need the surface area of the tank: the total area of all its outside surfaces. The tank is a cylinder, and a cylinder has a neat secret.

What a cylinder unfolds into

A cylinder has two equal circles (the top and the base) and one curved surface. Think of a tin of tomato paste. Peel off its paper label and lay it flat: the label is a rectangle.

The rectangle's length is the distance round the tin: the circumference, 2 × π × r.
The rectangle's width is the height of the tin, h.

The formulas

Area of the two circles = 2 × π × r × r
Area of the curved surface = 2 × π × r × h
Total surface area = 2 × π × r × r + 2 × π × r × h

Worked example 1

a tin with radius 4 cm and height 10 cm.

Two circles: 2 × 3.14 × 4 × 4 = 2 × 3.14 × 16 = 100.48 cm².
Curved surface: 2 × 3.14 × 4 × 10 = 251.2 cm².
Total: 100.48 + 251.2 = 351.68 cm².

Worked example 2

an open tank with no lid. Radius 1 m, height 2 m.

Only one circle, the base: 3.14 × 1 × 1 = 3.14 m².
Curved surface: 2 × 3.14 × 1 × 2 = 12.56 m².
Total: 3.14 + 12.56 = 15.7 m².

At 5 m² per tin, Niyonzima's family needs 15.7 ÷ 5 = 3.14 tins, so they must buy 4 tins.

Worked example 3

(backwards): the label of a tin has an area of 188.4 cm², and the tin has a radius of 3 cm. How tall is it?

Label area = 2 × π × r × h, so 188.4 = 2 × 3.14 × 3 × h = 18.84 × h.
h = 188.4 ÷ 18.84 = 10 cm.

Tip

Label, lid and base. The label is a rectangle (circumference × height). The lid and base are two circles (π × r × r each). Leave out any circle that is not there, like the lid of an open tank.

✦

Try it yourself

Use π = 3.14.

Find the curved surface area of a cylinder with radius 2 cm and height 5 cm.
Find the total surface area of the same cylinder.
A pipe (open at both ends) has a radius of 0.5 m and is 4 m long. What is the area of its curved surface?
For the tin in worked example 1, which is bigger: the label or the two circles together?

Answers

2 × 3.14 × 2 × 5 = 62.8 cm².
Two circles: 2 × 3.14 × 2 × 2 = 25.12 cm². Total: 25.12 + 62.8 = 87.92 cm².
2 × 3.14 × 0.5 × 4 = 12.56 m².
The label: 251.2 cm² is bigger than 100.48 cm².

Quick check

Use π = 3.14. A cylinder has a radius of 5 cm and a height of 10 cm. What is its curved surface area?

What you learned today

A cylinder's net is two circles and one rectangle.
The rectangle is 2 × π × r long and h wide, so the curved surface area is 2 × π × r × h.
Total surface area = 2 × π × r × r + 2 × π × r × h; count only the circles that are really there.

Sources and credits

Prealgebra 2e, 9.7 Solve Geometry Applications: Volume and Surface Area by OpenStax, via LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.

The water-tank story, the cylinder net drawing and the worked examples were written by Ganzaa.

1What a cylinder unfolds into

2The formulas

Try it yourself

What a cylinder unfolds into

The formulas