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Area, volume and compound unit conversions

Apply unit factors to every length dimension and convert the numerator and denominator of compound units independently.

  • SEC G3 Physics 2027

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A centimetre-to-metre conversion applies once to a length, twice to an area and three times to a volume. Keep the whole unit factor in brackets before raising it to a power; for a compound unit, convert both its numerator and denominator.

Apply the factor to every dimension

The prefix equivalence is 1 cm = 10⁻² m. For a square of side 1 cm, both sides are 0.01 m; for a cube, all three edges are 0.01 m.

Apply the length conversion to every dimensionA square with two sides of 1 centimetre has area (0.01 metre) squared, or 10 to the power minus 4 square metres. A cube with three edges of 1 centimetre has volume (0.01 metre) cubed, or 10 to the power minus 6 cubic metres.Area: two dimensionsVolume: three dimensions1 cm = 0.01 m1 cm = 0.01 m1 cm = 0.01 m1 cm1 cm1 cm² = (0.01 m)²= 10⁻⁴ m²1 cm³ = (0.01 m)³= 10⁻⁶ m³
Scroll across the figure to read all labels.
Convert both dimensions of an area, or all three dimensions of a volume. These shapes are schematic and are not drawn to a shared physical scale.
1 cm² = (10⁻² m)² = 10⁻⁴ m²
1 cm³ = (10⁻² m)³ = 10⁻⁶ m³

The square and cube stay the same size when described in metres. Only the numbers and unit symbols change. A cubic centimetre is one-millionth of a cubic metre, so a given volume needs a much smaller numerical value in m³ than in cm³.

Reverse the direction using the same equivalence

Since 1 m = 100 cm:

1 m² = 10⁴ cm², 1 m³ = 10⁶ cm³

Do not decide the factor from whether the number “looks too big”. Write the equivalence and check whether the new unit is larger or smaller than the old one.

Treat compound units as a fraction

Density has units of mass divided by volume. To convert g cm⁻³ to kg m⁻³, convert grams and cubic centimetres:

1 g cm⁻³ = (10⁻³ kg)/(10⁻⁶ m³) = 10³ kg m⁻³

The negative power means “per cubic centimetre”. Converting the volume in the denominator therefore involves division by 10⁻⁶, not multiplication by it. Mass and volume unit changes work together; the material’s density has not changed.

Only convert when needed. A density calculation using grams and cubic centimetres directly gives g cm⁻³; those are suitable consistent units if that is the requested answer.

Work through a cubic and a compound conversion

Guided practice 1

Volume conversion

About 5 min

Problem

Convert 40.0 cm ³ to cubic metres.

Apply the conversion to all three dimensions

Unit: m^3

Hints

Hint 1: bracket the length conversion

Start with 1 cm = 10⁻² m.

Hint 2: cube the whole factor

Use (10⁻²)³ = 10⁻⁶.

Show solution step by step

  1. Convert the cubic unit

    Method

    Cube the centimetre conversion factor.

    Reason

    A volume contains three length dimensions.

    Working

    40.0 cm ³ = 40.0 × (10⁻² m)³

  2. Simplify in standard form

    Method

    Combine the coefficient with 10⁻⁶.

    Reason

    The requested unit is cubic metres.

    Working

    40.0 × 10⁻⁶ m ³ = 4.00 × 10⁻⁵ m ³

Try it yourself 2

Density conversion

Minimal support

Problem

Convert 7.8 g cm⁻³ to kg m⁻³.

Convert numerator and denominator separately

Hints

Hint 1: convert mass

Use 1 g = 10⁻³ kg.

Hint 2: convert cubic centimetres

Use 1 cm ³ = 10⁻⁶ m ³ in the denominator.

Show solution step by step

  1. Replace both unit factors

    Method

    Convert grams in the numerator and cubic centimetres in the denominator.

    Reason

    Density is a compound unit, so both parts must become consistent.

    Working

    7.8 g cm⁻³ = 7.8 × (10⁻³ kg)/(10⁻⁶ m³)

  2. Combine the powers

    Method

    Divide the two powers of ten.

    Reason

    10⁻³/10⁻⁶ = 10³.

    Working

    7.8 × 10³ kg m⁻³

Try without prompts

Check your understanding 1: Convert an area

A square has side 3.2 cm. Calculate its area in m².

Show answer

Convert the length before squaring:

3.2 cm = 3.2 × 10⁻² m

A = (3.2 × 10⁻²)² = 1.024 × 10⁻³ m²

To two significant figures, A = 1.0 × 10⁻³ m².

Mind stretcher 1: Convert both measurements and check the resultExtension

A sample has mass 18.0 g and volume 6.00 cm³. Calculate its density in g cm⁻³, then obtain the density in kg m⁻³ in two ways: by converting the density, and by converting the mass and volume before dividing. Check that the answers agree.

Show answer

The density is 18.0/6.00 = 3.00 g cm⁻³. Multiplying by 10³ gives 3.00 × 10³ kg m⁻³.

Alternatively, 18.0 g = 0.0180 kg and 6.00 cm³ = 6.00 × 10⁻⁶ m³. Dividing those converted quantities also gives 3.00 × 10³ kg m⁻³. The agreement checks that both unit changes were applied correctly.

Keep the factor with its unit

Use a squared factor for area and a cubed factor for volume. In a compound unit, keep numerator and denominator conversions separate until you combine their powers. The physical quantity stays unchanged throughout.

Syllabus and review details