Lesson 6.6 · 35 min

Area and Volume Units, and Similar Solids

Converting squared and cubed units, capacity, and the length, area and volume scale factors of similar shapes and solids — including working backwards.

In this lesson you will learn to…

Convert between mm², cm², m², hectares and km², and between mm³, cm³, m³ and litres
Put every length in the same unit before calculating, including in multi-step problems
Use the length, area and volume scale factors $k$ , $k^{2}$ and $k^{3}$ of similar shapes
Work backwards from an area or volume ratio to a length ratio
Solve problems with similar solids, such as finding a mass or capacity

Squared and cubed units

A square metre is $100$ cm by $100$ cm, so $1\ \text{m}^2 = 10\,000\ \text{cm}^2$ — not 100. A cubic metre is $100 \times 100 \times 100 = 1\,000\,000\ \text{cm}^3$ . Whatever the length conversion is, square it for area and cube it for volume.

Try it

Why units square and cube

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Units

Along one edge: 1 m = 100 cm
Volume: 1 m³ = 100 × 100 × 100 = 1 000 000 cm³
m³ → cm³: × 1 000 000
cm³ → m³: ÷ 1 000 000
Capacity link: 1 m³ = 1000 L
Note: Drawn with 10 strips each way — really 100

Switch between Length, Area and Volume, and between the unit pairs. The conversion factor is multiplied once, twice or three times — the same reason similar solids scale by $k$ , $k^{2}$ and $k^{3}$ .

Length	Area	Volume
$1\ \text{cm} = 10\ \text{mm}$	$1\ \text{cm}^2 = 100\ \text{mm}^2$	$1\ \text{cm}^3 = 1000\ \text{mm}^3 = 1$ mL
$1\ \text{m} = 100\ \text{cm}$	$1\ \text{m}^2 = 10\,000\ \text{cm}^2$	$1\ \text{m}^3 = 1\,000\,000\ \text{cm}^3 = 1000$ L
$1\ \text{km} = 1000\ \text{m}$	$1\ \text{km}^2 = 1\,000\,000\ \text{m}^2 = 100$ ha	—

In a problem with mixed units, convert every length first, then calculate. Converting a finished area or volume is possible but needs the squared or cubed factor.

Similar solids

Two solids are similar if one is an enlargement of the other. If every length is multiplied by $k$ :

every area (a face, the surface area, a cross-section) is multiplied by $k^2$ ;
every volume — and so every capacity, and every mass if the material is the same — is multiplied by $k^3$ .

Try it

Enlarging a solid

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Scale factor, k1

Scale factor: k = 1
Lengths: × 1
Areas: × 1² = × 1
Volumes: × 1³ = × 1

Set $k = 3$ . How many copies of the original fit inside? Now switch to Area: how many copies of one face fit on the enlarged face?

Working backwards

Often you are given an area ratio or a volume ratio and asked for something else. The route always goes through the lengths:

$\text{area ratio} \xrightarrow{\sqrt{\ }} \text{length ratio} \xrightarrow{(\ )^3} \text{volume ratio}$

Key rules and formulas

Converting squared and cubed units
$1 m^{2} = 10 0^{2} cm^{2} = 10000 cm^{2} 1 m^{3} = 10 0^{3} cm^{3} = 1000000 cm^{3}$
Square the length conversion for area, cube it for volume. Also: $1$ ha $= 10000 m^{2}$ , $1 km^{2} = 100$ ha, $1 cm^{3} = 1$ mL, $1 m^{3} = 1000$ L.
Similar shapes and solids
$lengths \times k areas \times k^{2} volumes (and masses, capacities) \times k^{3}$
$k$ is the length scale factor, e.g. $\frac{new height}{old height}$ . Mass and capacity behave like volume when the material is the same.
Working backwards
Given an area ratio, square root it to get the length ratio. Given a volume ratio, cube root it. Always go back to lengths before going forward to the other measure.

Worked examples

Worked example 1

Rain on a roof (mixed units)

A roof is

12 m

8.5 m

18 mm

of rain falls on it. How many litres is that?

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

Worked example 2

Similar solids: area to volume

Two similar jugs have surface areas

180 cm^{2}

and

405 cm^{2}

. The smaller holds

0.8

litres. How much does the larger hold?

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

Worked example 3

Similar solids: mass to length

Two similar solid metal cones have masses

320

g and

40

g. The larger has radius

6

cm. Find the radius of the smaller.

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

Practice: area and volume units, and similar solids

12 questions

Convert every length first. For similar solids, always go back to the length scale factor.

Question 1 of 12Sort into groups
Sort each conversion by the factor you multiply or divide by.
Tap a card, then tap the group it belongs in. You can also drag.
Question 2 of 12Type the answer
Convert $0.0375 m^{2}$ to $mm^{2}$ .
Your answermm²
Question 3 of 12Type the answer
A field is $0.36 km^{2}$ . How many hectares is that?
Your answerha
Question 4 of 12Type the answerChallenge
A swimming pool is $25 m$ long and $12 m$ wide. The water is $1.2 m$ deep at one end and $2 m$ deep at the other, sloping evenly. How many litres of water does it hold?
Your answerL
Question 5 of 12Step by stepChallenge
Mixed units in a cylinder.
A cylindrical water tank has diameter $1.4 m$ and height $95 cm$ . Water flows in at $12$ litres per minute. Use $π = 3.142$ .
1. Find the volume of the tank in $m^{3}$ , to 3 significant figures.
  Step 1 answerm³

Question 6 of 12Complete the table

Two solids are similar with length scale factor

k

. Complete the table.

Length factor $k$	Area factor	Volume factor
$3$	Row 1, Area factor	Row 1, Volume factor
Row 2, Length factor k	$25$	Row 2, Volume factor
Row 3, Length factor k	Row 3, Area factor	$64$
$\frac{1}{2}$	Row 4, Area factor	Row 4, Volume factor

Question 7 of 12Type the answer
Two similar cones have heights $8$ cm and $12$ cm. The smaller has volume $96 cm^{3}$ . Find the volume of the larger.
Your answercm³
Question 8 of 12Type the answerChallenge
Two similar containers have capacities $1.6$ litres and $5.4$ litres. The smaller has a base area of $80 cm^{2}$ . Find the base area of the larger.
Your answercm²
Question 9 of 12Type the answer
Two similar solid statues are made of the same bronze. Their heights are $15$ cm and $45$ cm. The small one has mass $0.6$ kg. Find the mass of the large one.
Your answerkg
Question 10 of 12Type the answerChallenge
Two similar cylinders have surface areas $72 cm^{2}$ and $200 cm^{2}$ . The larger has volume $250 cm^{3}$ . Find the volume of the smaller.
Your answercm³
Question 11 of 12Multiple choiceChallenge
A model car is made to a scale of $1 : 40$ . The real car's windscreen has area $1.2 m^{2}$ . What is the area of the model's windscreen?
Question 12 of 12Explore and match
Two similar boxes have volumes in the ratio $1 : 64$ . Set the length scale factor $k$ .
Move the controls until you match the question, then press Check.
Show
Scale factor, k1
Scale factor
k = 1
Lengths
× 1
Areas
× 1² = × 1
Volumes
× 1³ = × 1
Aim for
volume × 64

In this lesson you will learn to…

Squared and cubed units

Why units square and cube

Similar solids

Enlarging a solid

Working backwards

Key rules and formulas

Converting squared and cubed units

Similar shapes and solids

Working backwards

Worked examples

Practice: area and volume units, and similar solids