To find a volume from mass and density, use V = m ÷ ρ. Keep the units matched: kg with kg/m³ gives m³, and g with g/cm³ gives cm³.
This lesson is part of mass, weight and density. It follows density unit conversion, which you will need again here.
How do you rearrange the density formula?
Density is mass per unit volume: ρ = m/V. To make V the subject, multiply both sides by V and then divide both sides by ρ. That gives V = m/ρ.
A triangle with m on top and ρ and V underneath is a common memory aid. Covering the unknown shows the formula, but check the units anyway.
Step by step
- List the data with units, and mark the unknown.
- Match the units. Convert so that mass and density use the same system.
- Write V = m/ρ, substitute and calculate.
- Give the unit and, if useful, convert to a practical one.
- Sense check: a denser material takes up less space for the same mass.
Worked example
A solid aluminium bar has a mass of 0.54 kg. The density of aluminium is 2700 kg/m³. Find the volume of the bar in m³ and in cm³. (Invented data.)
Step 1: data: m = 0.54 kg, ρ = 2700 kg/m³, V = ?
Step 2: units already match (kg and kg/m³).
Step 3: V = m/ρ = 0.54 ÷ 2700 = 0.000 2 m³ = 2.0 × 10⁻⁴ m³.
Step 4: in cm³, multiply by 1 000 000: 0.000 2 × 1 000 000 = 200 cm³.
Check: in g and g/cm³: 540 g ÷ 2.7 g/cm³ = 200 cm³. The second route gives the same answer.
The mistake to watch for
Mistaken working: V = ρ/m = 2700 ÷ 0.54 = 5000
The student flipped the formula. The number 5000 has no sensible unit.
The units expose the error: (kg/m³) ÷ kg gives 1/m³, which is not a volume. A second warning sign is size.
A volume of 5000 m³ for a small bar is unreasonable. Always check that the unit of your answer is a volume.
Check yourself
1. An oil has density 0.85 g/cm³. Find the volume of 850 g of it.
Show answer
V = m/ρ = 850 ÷ 0.85 = 1000 cm³.
2. What volume does 4.0 kg of water occupy? Use 1000 kg/m³. Give the answer in m³ and in cm³.
Show answer
V = 4.0 ÷ 1000 = 0.004 m³. In cm³: 0.004 × 1 000 000 = 4000 cm³ (that is 4 litres).
3. A gold ingot has a mass of 386 g. Gold has density 19.3 g/cm³. Find the volume.
Show answer
V = 386 ÷ 19.3 = 20 cm³. Check: 19.3 × 20 = 386.
Where this leads next
Volume and density together decide whether something floats, so continue to explaining floating with an appropriate force model. For practice on the whole chain, use the mixed practice set.
If you can do the algebra but unit matching keeps slipping, a teacher can build a short routine for it in online one-to-one Physics tuition.