Density measures how much mass is packed into a unit of volume. The formula is Density = Mass ÷ Volume (D = M/V). At GCSE, problems combine this formula with volume calculations for cylinders, spheres, cones and prisms, and may require converting between g/cm³ and kg/m³.
The density formula and its rearrangements
The three forms of the density equation are:
| Want to find | Formula | Units example |
|---|---|---|
| Density (D) | D = M ÷ V | g/cm³ |
| Mass (M) | M = D × V | g |
| Volume (V) | V = M ÷ D | cm³ |
Memory aid — formula triangle: Write M at the top; D and V side by side at the bottom. Cover the quantity you want: M alone means D × V; D alone means M ÷ V; V alone means M ÷ D.
The unit of density combines a mass unit and a volume unit, always written as mass per volume: g/cm³, kg/m³ or kg/litre.
Worked example 1 — finding mass from a cylinder
A steel cylinder has radius 4 cm and height 10 cm. The density of steel is 7.85 g/cm³. Find the mass of the cylinder.
Step 1 — Find the volume:
V = πr²h = π × 4² × 10 = 160π cm³ ≈ 502.65 cm³
Step 2 — Find the mass:
M = D × V = 7.85 × 502.65 ≈ 3946 g (to 4 significant figures)
= 3.946 kg (to 4 s.f.)
Worked example 2 — finding volume from density and mass
A gold nugget has mass 386 g. The density of gold is 19.3 g/cm³. Find the volume of the nugget.
V = M ÷ D = 386 ÷ 19.3 ≈ 20.0 cm³ (to 3 s.f.)
Worked example 3 — sphere
A lead sphere has radius 3 cm. The density of lead is 11.34 g/cm³. Find the mass.
Volume of sphere: V = (4/3)πr³ = (4/3) × π × 27 = 36π ≈ 113.1 cm³
Mass: M = D × V = 11.34 × 113.1 ≈ 1282 g (to 4 s.f.)
How do you convert between g/cm³ and kg/m³?
This conversion catches students out at GCSE because it involves both a mass conversion and a volume conversion.
- 1 kg = 1000 g
- 1 m = 100 cm, so 1 m³ = 100³ cm³ = 1,000,000 cm³
Therefore: 1 g/cm³ = (1/1000) kg / (1/1,000,000) m³ = 1000 kg/m³
Conversion rule: 1 g/cm³ = 1000 kg/m³
To convert g/cm³ → kg/m³: multiply by 1000.
To convert kg/m³ → g/cm³: divide by 1000.
Worked example 4:
Convert 2.7 g/cm³ (density of aluminium) to kg/m³.
2.7 × 1000 = 2700 kg/m³
Worked example 5 — mixed units requiring conversion
An iron beam is a rectangular prism 2 m × 0.3 m × 0.1 m. The density of iron is 7874 kg/m³. Find the mass in kg.
Volume: V = 2 × 0.3 × 0.1 = 0.06 m³
Mass: M = D × V = 7874 × 0.06 = 472.4 kg
Frequently asked questions
What is the difference between mass and weight?
In everyday language, "weight" often means mass. Scientifically, mass (measured in kg or g) is the amount of matter in an object; weight (measured in Newtons) is the gravitational force acting on it. In GCSE maths, density problems use mass, not weight. Do not convert to Newtons in a maths paper.
What happens to density when the same material is melted and recast into a different shape?
The density stays the same — it is a property of the material, not the shape. The volume and mass stay the same too. If the question says a shape is melted and recast, calculate the volume of the original shape and use that volume for the new shape.
What if a composite object is made of two different materials?
Find the volume of each part separately, calculate the mass of each using its own density, and add the masses. The total mass of the object is the sum; the overall density is total mass ÷ total volume — which will be somewhere between the two individual densities.
Can density be less than 1 g/cm³?
Yes. Ice has a density of approximately 0.92 g/cm³, which is why it floats on water (density 1 g/cm³). Objects with density less than water float; objects with density greater than water sink. This physical context appears in GCSE problems asking you to predict whether an object will float.
Need help with density calculations? Professor Pi works through each step at aitutors.me.