Length, Area and Volume Units

Length measures along one dimension, area over two dimensions, and volume through three. Their units carry that structure. A square metre is not a longer kind of metre, and a cubic metre is not simply a larger square metre.

The familiar formulas for rectangles and boxes make the conversion rules visible. This lesson applies those rules to capacity, material quantities and measurements made in different units. The general shape formulas remain in Area and Perimeter and Surface Areas and Volumes.

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Count unit lengths, unit squares and unit cubes

A 1 m segment contains 100 lengths of 1 cm. A square of side 1 m contains 100 rows of 100 square centimetres. A cube of side 1 m contains 100 layers of those arrays:

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.

The conversion factor is raised to the same power as the unit. More generally, if one length unit contains $k$ of another length unit, the corresponding area and volume factors are $k^2$ and $k^3$ .

Convert areas without losing a dimension

Convert 2500 square centimetres to square metres:

2500\text{ cm}^2\left(\frac{1\text{ m}}{100\text{ cm}}\right)^2=0.25\text{ m}^2.

The square on the factor applies to both the number and the unit. Dividing by 100 instead of 10 000 would convert only one dimension.

A hectare is 10 000 square metres. Therefore 0.35 ha represents 3500 square metres. A hectare does not specify the shape of the region; a long narrow field and a square field can have equal area but different perimeters.

For a 4.8 m by 3.6 m room,

A=4.8\times3.6=17.28\text{ m}^2,\qquad P=2(4.8+3.6)=16.8\text{ m}.

These numbers describe different quantities. Flooring calculations use area; skirting or edge trim uses the relevant boundary length, with openings handled separately.

Connect litres to cubic units

The NIST volume reference identifies the useful exact relationships

1\text{ L}=1\text{ dm}^3=1000\text{ cm}^3,

1\text{ mL}=1\text{ cm}^3,\qquad1\text{ m}^3=1000\text{ L}.

An internal tank length of 80 cm, width of 50 cm and height of 40 cm gives

V=80\times50\times40=160\,000\text{ cm}^3=160\text{ L}.

Using metres instead gives $0.8\times0.5\times0.4=0.16$ cubic metres, again 160 L. Two independent unit routes help expose a missed factor of 1000.

Capacity is not the same as the current contents

The drawing is a side view, not a three-dimensional scale model. All listed dimensions are internal.

Diagram explanation and exact data

The base area is 80 × 50 = 4000 square centimetres. Each centimetre of depth adds 4000 cubic centimetres, or four litres. At depth 25 cm there are 100 litres; depth 40 cm gives 160 litres.

Capacity is not the same as the current contents: exact reference data
Depth (cm)	Volume (cm³)	Volume (L)
0	0	0
25	100000	100
40	160000	160

Separate internal capacity from external volume

For an open rectangular box with internal dimensions 20 cm by 30 cm by 40 cm, the internal capacity is

20\times30\times40=24\,000\text{ cm}^3=24\text{ L}.

If its external dimensions are 22 cm by 32 cm by 41 cm, its exterior bounding volume is $28\,864\text{ cm}^3$ . That does not make its capacity 28.864 L. The bounding volume includes the walls and base.

Similarly, the capacity of the 160 L tank is not necessarily its current liquid volume. At a liquid depth of 25 cm, a constant rectangular cross-section gives

V=80\times50\times25=100\,000\text{ cm}^3=100\text{ L}.

The capacity calculation assumes the full stated interior is usable. An operating fill limit may be lower and must be supplied separately.

Calculate material quantities before rounding to packages

A coating is specified, for this teaching example, to cover 8 square metres per litre per coat. Applying two coats to 17.28 square metres requires

V=\frac{17.28\text{ m}^2\times2}{8\text{ m}^2/\text{L}}=4.32\text{ L}.

If only 1 L containers can be bought, at least five are needed for this stated coverage model. Buying five litres does not mean the surface mathematically requires five litres; packaging rounds the purchase quantity upwards. Waste, absorption and manufacturer instructions are separate inputs, not invisible constants.

For soil spread over 12 square metres to a depth of 5 cm, convert the depth to 0.05 m first:

V=12\text{ m}^2\times0.05\text{ m}=0.60\text{ m}^3=600\text{ L}.

A mass estimate would additionally require bulk density. Volume alone does not determine how many kilograms to order.

Distinguish a unit conversion from a resized object

Changing the tank dimensions from centimetres to metres leaves the tank unchanged. Halving all three actual dimensions creates a different tank with $(1/2)^3=1/8$ of the volume: 20 L rather than 80 L.

This contrast separates unit conversion from geometric scaling. The first changes the numerical representation of one quantity. The second changes the object itself.

Practice questions

Convert 0.06 square metres to square centimetres.
A box has internal dimensions 40 cm by 30 cm by 25 cm. Find its capacity in litres.
A 10 square-metre layer is 8 cm deep. Find its volume in cubic metres and litres.

Check your answers

$0.06\times10\,000=600\text{ cm}^2$ .
$40\times30\times25=30\,000\text{ cm}^3=30\text{ L}$ .
The depth is 0.08 m, so $V=10\times0.08=0.8\text{ m}^3=800\text{ L}$ .