Lesson 6.4 · 22 min

Volume of Prisms and Cylinders

One formula for every prism — cross-section times length — and what to do when you are given the volume instead.

In this lesson you will learn to…

Find the volume of any prism using cross-section × length
Find the volume of a cylinder
Work backwards from a volume to find a missing dimension
Convert between $cm^{3}$ , litres and $m^{3}$

One formula for all of them

Every prism has the same volume formula:

$V = \text{area of cross-section} \times \text{length}$

That is genuinely all of it. The reason it works: a prism is the cross-section “extruded” — pushed along in a straight line. Stack $h$ layers, each one unit thick and each of area $A$ , and you have $A \times h$ .

Prism	Cross-section	Volume
Cuboid	rectangle, $l \times w$	$l \times w \times h$
Triangular prism	triangle, $\frac{1}{2}bh$	$\frac{1}{2}bh \times \text{length}$
Cylinder	circle, $\pi r^2$	$\pi r^2 h$

So there is only one formula to learn, plus the area formulas you already know.

The method

Identify the cross-section — the face that repeats all the way along.
Find its area.
Multiply by the length between the two ends.
Use cubed units.

Step 1 is the one worth slowing down for. On a triangular prism lying on its side, the cross-section is the triangle, not the rectangle underneath it — so the “length” you multiply by is the horizontal distance, not the vertical height of the triangle.

Working backwards

If the question gives you the volume and asks for a missing dimension, put everything you know into the formula and solve the equation:

$V = \pi r^2 h \quad\Rightarrow\quad r^2 = \frac{V}{\pi h} \quad\Rightarrow\quad r = \sqrt{\frac{V}{\pi h}}$

As with circles, divide first and square root last.

Units: the bit that catches everyone

$1\ \text{cm}^3 = 1\ \text{ml} \qquad 1000\ \text{cm}^3 = 1\ \text{litre} \qquad 1\ \text{m}^3 = 1\,000\,000\ \text{cm}^3$

That last one surprises people. A metre is 100 cm, but a cubic metre is $100 \times 100 \times 100 = 1\,000\,000$ cubic centimetres, because all three dimensions scale.

The same logic applies to areas: $1\ \text{m}^2 = 10\,000\ \text{cm}^2$ , not 100.

A quick sense check. Volumes get big fast. A cylinder with radius 3 cm and height 10 cm holds about 283 ml — roughly a small can of drink. If your answer comes out as 28 ml or 2830 ml, check your working.

Key rules and formulas

Volume of any prism
$V = A \times h$
where $A$ is the area of the cross-section and $h$ is the length (or height) between the two identical ends. This single formula covers cuboids, triangular prisms and cylinders.
Volume of a cylinder
$V = π r^{2} h$
Just $A \times h$ with $A = π r^{2}$ . Nothing new to remember.
Working backwards
Given the volume and asked for a dimension, substitute what you know and solve. For a cylinder: $r = \frac{V}{π h}$ .
Volume units
$1 cm^{3} = 1 ml 1000 cm^{3} = 1 litre 1 m^{3} = 1000000 cm^{3}$
Volume is always cubed: $cm^{3}$ , $m^{3}$ . The jump from cm to m is $10 0^{3} = 1000000$ , not 100 — this catches people out constantly.

Try it

Volume and surface area together

Length (cm)5

Width (cm)4

Height (cm)3

Volume = l × w × h: 5 × 4 × 3 = 60 cm³
Surface area = 2(lw + lh + wh): 94 cm²

Change the dimensions and watch both numbers. Try making a long thin box and a near-cube with a similar volume — the long thin one has far more surface area. That is why packaging tends towards cubes.

Worked examples

Worked example 1

A triangular prism

A prism has a triangular cross-section of area

60 m^{2}

and is

15 m

long. Find its volume.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A cylinder

Find the volume of a cylinder with radius

3 cm

and height

10 cm

. Use

π = 3.14

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Working backwards

A cylinder has volume

1570 cm^{3}

and height

20 cm

. Find its radius. Use

π = 3.14

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

Volume to litres

A tank measures

50 cm \times 40 cm \times 30 cm

. How many litres does it hold?

Try it yourself first, then reveal the solution one step at a time.

Practice: volume

9 questions

Nine questions, including one where you build a cuboid to a target volume.

Question 1 of 9Type the answerFoundation
A prism has a cross-section of area $60 m^{2}$ and is $15 m$ long. Find its volume.
Your answerm³
Question 2 of 9Type the answer
Find the volume of a cylinder with radius $3 cm$ and height $10 cm$ . Use $π = 3.14$ .
Your answercm³
Question 3 of 9Explore and match
Set the sliders to build a cuboid with volume exactly $60 cm^{3}$ , using length $5$ , width $4$ and height $3$ .
Move the controls until you match the question, then press Check.
Length (cm)6
Width (cm)3
Height (cm)4
Volume = l × w × h
6 × 3 × 4 = 72 cm³
Surface area = 2(lw + lh + wh)
108 cm²
Question 4 of 9Step by stepChallenge
Work backwards from a volume.
A cylinder has volume $1570 cm^{3}$ and height $20 cm$ . Use $π = 3.14$ .
1. First work out $π \times h$ , that is $3.14 \times 20$ .
  Step 1 answer
Question 5 of 9Type the answer
A tank measures $50 cm \times 40 cm \times 30 cm$ . How many litres does it hold?
Your answerlitres
Question 6 of 9Match the pairs
Match each prism to its volume formula.
Pick an answer, then choose the row it belongs to. You can also drag it.
- Cuboid
- Cylinder
- Triangular prism
- Any prism
Question 7 of 9Fill the gapsChallenge
Complete these volume conversions.
$1 cm^{3} =$ Gap ml ml. $1$ litre $=$ Gap litre $cm^{3}$ . $1 m^{3} =$ Gap m3 $cm^{3}$ .
Question 8 of 9Multiple choice
A triangular prism has a cross-section with base $8 cm$ and height $5 cm$ , and it is $12 cm$ long. What is its volume?

Question 9 of 9Complete the table

Complete the table for cylinders. Use

π = 3.14

and round volumes to 1 decimal place.

Radius (cm)	Height (cm)	Cross-section area ( $cm^{2}$ )	Volume ( $cm^{3}$ )
2	5	Row 1, Cross-section area (cm2)	Row 1, Volume (cm3)
5	4	Row 2, Cross-section area (cm2)	Row 2, Volume (cm3)
10	2	Row 3, Cross-section area (cm2)	Row 3, Volume (cm3)

In this lesson you will learn to…

One formula for all of them

The method

Working backwards

Units: the bit that catches everyone

Key rules and formulas

Volume of any prism

Volume of a cylinder

Working backwards

Volume units

Volume and surface area together

Worked examples

Practice: volume