Volume • Topic 3 of 3

Volume Word Problems with Missing Dimensions

Many real problems give you the volume and ask for a length you cannot see — the depth of a pool, the height of a carton, the width of a tank. The same formula handles every case once you know how to rearrange it.

The four problem types

Type	What is given	What to do
1. Find the volume	$l$, $w$, $h$	Multiply: $V = l \times w \times h$
2. Find a missing dimension	$V$ and two of $l, w, h$	Divide: $V \div (\text{product of the two known})$
3. Compare volumes	two solids	Find both volumes, then compare
4. Multi-step / capacity	mixed	May need unit change ($1$ L $=1000$ cm$^3$)

Rearranging the formula. Starting from $V = l \times w \times h$:

height $= V \div (l \times w)$
width $= V \div (l \times h)$
length $= V \div (w \times h)$

In words: the missing dimension equals the volume divided by the product of the two dimensions you do know.

A 4-step method that always works:

Read the problem and note the units.
Write $V = l \times w \times h$ and fill in what you know.
Solve — multiply to find a volume, or divide to find a missing dimension.
Check the unit (cubic for volume, single for a length) and that the answer is sensible.

Capacity link. Liquids are measured in litres, but volume of a container is in cm$^3$. Remember $1$ litre $= 1000$ cm$^3$, so a tank of $5000$ cm$^3$ holds $5$ litres.

STRATEGY FOR VOLUME WORD PROBLEMS:

   1. Write the formula:  V = l x w x h
   2. Put in the numbers you KNOW.
   3. If a dimension is MISSING, divide:
          missing = V / (product of the two known dimensions)
   4. Check the unit is cubic (cm3, m3, ...).

MISSING-DIMENSION EXAMPLE:  V = 120, l = 5, w = 4, find h

   120 = 5 x 4 x h
   120 = 20 x h
   h = 120 / 20 = 6

CAPACITY LINK:  1 litre = 1000 cm3  (so a 2 L jug holds 2000 cm3)

Solved Examples

Worked Example

A shipping container is 12 m long, 8 m wide and 6 m tall. Find its volume.

Solution

$V = 12 \times 8 \times 6$
$V = 96 \times 6 = 576$
Answer: $576$ m$^3$

Worked Example

A rectangular prism has volume 240 cm$^3$. Its length is 10 cm and width is 6 cm. Find its height.

Solution

height $= V \div (l \times w)$
$h = 240 \div (10 \times 6) = 240 \div 60$
$h = 4$
Answer: $4$ cm

Worked Example

Box A is $8 \times 5 \times 4$ cm and Box B is $6 \times 6 \times 5$ cm. Which box has the greater volume?

Solution

Box A: $V = 8 \times 5 \times 4 = 160$ cm$^3$
Box B: $V = 6 \times 6 \times 5 = 180$ cm$^3$
$180 > 160$
Answer: Box B

Worked Example

A rectangular pool is 20 m long and 10 m wide. Its volume of water is 600 m$^3$. How deep is the pool?

Solution

depth $= V \div (l \times w)$
$= 600 \div (20 \times 10) = 600 \div 200$
$= 3$
Answer: $3$ m deep

Worked Example

A juice carton holds 2 litres. Its base is 10 cm by 10 cm. Find its height. (Use $1$ L $=1000$ cm$^3$.)

Solution

$2$ L $= 2000$ cm$^3$
height $= V \div (l \times w) = 2000 \div (10 \times 10)$
$= 2000 \div 100 = 20$
Answer: $20$ cm

Worked Example

How many 2-cm cubes fit inside a box that is 8 cm by 6 cm by 4 cm?

Solution

Box volume $= 8 \times 6 \times 4 = 192$ cm$^3$
Each small cube $= 2 \times 2 \times 2 = 8$ cm$^3$
Number of cubes $= 192 \div 8 = 24$
Answer: $24$ cubes

Key Points

Always start by writing $V = l \times w \times h$.
Missing dimension = volume $\div$ (product of the two known dimensions).
To compare solids, work out both volumes first, then compare.
Capacity: $1$ litre $= 1000$ cm$^3$ — convert before dividing.
Check your unit: a volume is cubic; a single length is not.
A quick check: multiply your answer back and see if you get the original volume.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.To find a missing height, you compute:

Explanation: Divide $V$ by the base.

Q2.A prism has $V=240$, $l=10$, $w=6$. Its height is:

Explanation: $240\div60=4$.

Q3.$1$ litre equals:

Explanation: $1$ L $=1000$ cm$^3$.

Q4.Box A $=2\times3\times4$, Box B $=3\times3\times3$. Which is bigger?

Explanation: $24<27$, so B.

Practice more MCQs → 📝 Assignment · 35 marks