Cylinders – GCSE Maths Boost

How to calculate the volume and surface area of a cylinder

A cylinder is a 3D shape with two equal circular ends and one curved surface, like a drinks can.

Surface area, volume and density - Geometry and trigonometry - UniSkills - Curtin Library

The radius $r$ r is the distance from the centre of the circle to its edge. The height $h$ is the distance between the two circular ends.

There are two main calculations you need to know:

Volume — how much space is inside the cylinder.
Surface area — the total area covering the outside of the cylinder.

1. How to calculate the volume

The formula is:

$V = π r^{2} h$ Where:

$V$ = volume
$π$ (pi) ≈ 3.14
$r$ = radius of the circular end
$h$ = height of the cylinder

Example

A cylinder has a radius of 3 cm and a height of 10 cm.
Step 1: Write the formula

$V = π r^{2} h$ Step 2: Substitute the values

$V = π \times 3^{2} \times 10$ Step 3: Square the radius

$V = π \times 9 \times 10$ Step 4: Multiply

$V = 90 π$ Using $π \approx 3.14$

$V \approx 90 \times 3.14$ Volume ≈ 282.7 cm³

Why does this work? The area of a circle is $π r^{2}$ πr2. Multiplying that area by the height gives the volume of the cylinder.

2. How to calculate the surface area

Surface area means the total area of every outside surface of the cylinder.

The formula is:

$S A = 2 π r^{2} + 2 π r h$ This formula has two parts:

$2 π r^{2}$ = the area of the two circular ends.
$2 π r h$ = the area of the curved surface.

Example

Use the same cylinder: radius = 3 cm and height = 10 cm.

Step 1: Write the formula

$S A = 2 π r^{2} + 2 π r h$ Step 2: Substitute the values

$S A = 2 π (3)^{2} + 2 π (3) (10)$ Step 3: Calculate each part

Two circular ends:

$2 π \times 9 = 18 π$ Curved surface:

$2 π \times 3 \times 10 = 60 π$ Step 4: Add the results

$S A = 18 π + 60 π = 78 π$

$S A \approx 78 \times 3.14159$ Surface area ≈ 245.0 cm²

3. Remember the difference

Volume	Surface area
$V = π r^{2} h$	$S A = 2 π r^{2} + 2 π r h$
Measures space inside	Measures the outside covering
Units are cubic, e.g. cm³	Units are square, e.g. cm²

Exam tip: If the question gives you the diameter instead of the radius, divide the diameter by 2 first.

For example, if the diameter is 8 cm:

$r = 8 \div 2 = 4 cm$

Cylinders

1 / 10

The diagram shows a cylindrical tank. The tank has a top and a bottom.
5 of these tanks are going to be painted.
Each tank has a diameter of 40cm and a height of 1.8m.
Each pot of paint can cover 4m².
How many pots of paint are needed to paint the 5 tanks?
pots.

2 / 10

A solid cylinder has a radius of 5m and a height of hm.
The total surface area of the cylinder is 165πm²
What is the value of h?
m

3 / 10

A solid cylinder has a radius of rm and a height of 12m.
The volume of the cylinder is 507π m³
What is the value of r?
{blank}m

4 / 10

A solid is formed by placing a half cylinder on a rectangular prism.
The solid has a width of 2m, a total height of 3m, and a length of 8m.
What is the volume of the solid? Give your answer correct to 3 significant figures.
m³

5 / 10

A solid cylinder is cut in half to form a semi-cylinder with a radius of 20cm and a length of 75cm.
What is the volume of the semi-cylinder? Give your answer correct to 3 significant figures.
cm³

What is the total surface area of the semi-cylinder? Give your answer correct to 3 significant figures.
cm³

6 / 10

A solid cylinder has a radius of 14cm and a height of 25cm.
What is the volume of the cylinder? Give your answer correct to 3 significant figures.
cm³

7 / 10

A solid cylinders has a radius of 6m and a height of 7m.
What is the total surface area of the cylinder? Give your answer in terms of π.
πm²

8 / 10

A solid cylinder has a diameter of 7cm and a height of 15cm.
What is the total surface area of the cylinder? Give your answer correct to 3 significant figures.
cm²

9 / 10

A cylinder has a diameter of 9cm and a height of 11cm.
What is the volume of the cylinder? Give your answer correct to 1 decimal place.
cm³

10 / 10

A cylinder has a radius of 5cm and a height of 12cm.
What is the volume of the cylinder? Give your answer in terms of π.
πcm³

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