Surface Area and Volume of a Cylinder

A cylinder is one of the most common 3D shapes -- think of cans, pipes, and columns. In this article you will learn every formula you need and work through three complete examples.

What is a cylinder?
Key dimensions
Net of a cylinder
Base area
Lateral (side) area
Total surface area
Volume
Worked example 1 -- Total surface area
Worked example 2 -- Volume
Worked example 3 -- Finding the radius
Summary of formulas
Practice exercises

What is a cylinder?

A cylinder is a 3D solid with:

Two parallel circular bases of equal size
A curved lateral surface connecting the bases

The bases are always congruent circles, and the lateral surface wraps around them like a label on a tin can.

Key dimensions

Every cylinder is described by two measurements:

Radius $r$ -- the radius of each circular base
Height $h$ -- the perpendicular distance between the two bases

```

┌─────────────┐

╱ r ╲

╱ ●─────── ╲ ● = centre

│ │

│ h │

│ │

╲ ╱

└─────────────┘

```

The diameter is $d = 2 r$ . If a problem gives you the diameter, remember to halve it to get $r$ .

Net of a cylinder

If you "unroll" a cylinder and lay it flat, you get its net:

Two circles (top and bottom bases)
One rectangle whose width equals the circumference of the base ( $2 π r$ ) and whose height equals $h$

```

┌──── 2πr ────┐

│ │

h │ (rectangle) │

│ │

└──────────────┘

○ (circle, radius r) -- top base

○ (circle, radius r) -- bottom base

```

Understanding the net makes it easy to see where every part of the surface area formula comes from.

Base area

Each base is a circle with radius $r$ :

S_{base} = π r^{2}

Since a cylinder has two bases, the combined base area is:

2 S_{base} = 2 π r^{2}

Lateral (side) area

The lateral surface, when unrolled, is a rectangle of width $2 π r$ and height $h$ :

S_{lateral} = 2 π r h

This is the area of the "label" that wraps around the cylinder.

Total surface area

The total surface area is the sum of both bases and the lateral surface:

S = 2 π r^{2} + 2 π r h

This can be factored neatly:

S = 2 π r (r + h)

Use $S = 2 π r (r + h)$ as your go-to formula for the total surface area of a cylinder.

Volume

The volume of a cylinder equals the base area multiplied by the height:

V = π r^{2} h

This makes intuitive sense: you are "stacking" circles of area $π r^{2}$ up to a height of $h$ .

Worked example 1 -- Total surface area

Problem: A cylinder has radius

r = 5 cm

and height

h = 12 cm

. Find its total surface area. Solution:

S = 2 π r (r + h)

S = 2 π \cdot 5 \cdot (5 + 12)

S = 2 π \cdot 5 \cdot 17

S = 170 π

S \approx 534.07 cm^{2}

Worked example 2 -- Volume

Problem: A cylindrical water tank has diameter

d = 1.2 m

and height

h = 2 m

. How many litres of water can it hold? Solution:

First, find the radius: $r = \frac{d}{2} = \frac{1.2}{2} = 0.6 m$ .

V = π r^{2} h = π \cdot 0. 6^{2} \cdot 2 = π \cdot 0.36 \cdot 2 = 0.72 π

V \approx 2.262 m^{3}

Convert to litres ( $1 m^{3} = 1000 l$ ):

V \approx 2262 l

The tank can hold approximately 2 262 litres of water.

Worked example 3 -- Finding the radius

Problem: A cylinder has volume

V = 1000 cm^{3}

and height

h = 8 cm

. Find the radius. Solution:

Start from the volume formula and solve for $r$ :

V = π r^{2} h

r^{2} = \frac{V}{π h} = \frac{1 000}{π \cdot 8} = \frac{1 000}{8 π} = \frac{125}{π}

r = \frac{125}{π} \approx 39.79

r \approx 6.31 cm

Summary of formulas

Quantity	Formula
Base area	$S_{base} = π r^{2}$
Lateral area	$S_{lateral} = 2 π r h$
Total surface area	$S = 2 π r (r + h)$
Volume	$V = π r^{2} h$

Practice exercises

Put your knowledge to the test:

Surface Area and Volume of a Cylinder – Formulas and Examples

Surface Area and Volume of a Cylinder

Table of contents

What is a cylinder?

Key dimensions

Net of a cylinder

Base area

Lateral (side) area

Total surface area

Volume

Worked example 1 -- Total surface area

Worked example 2 -- Volume

Worked example 3 -- Finding the radius

Summary of formulas

Practice exercises

Surface Area and Volume of a Cylinder – Formulas and Examples

Surface Area and Volume of a Cylinder

Table of contents

What is a cylinder?

Key dimensions

Net of a cylinder

Base area

Lateral (side) area

Total surface area

Volume

Worked example 1 -- Total surface area

Worked example 2 -- Volume

Worked example 3 -- Finding the radius

Summary of formulas

Related articles

Practice exercises