Density Calculations

Density is the mass per unit volume of a substance. It tells you how much matter is packed into a given space.

The equation

$\rho = \frac{m}{V}$

Symbol	Quantity	Unit
$\rho$	Density	kg/m³ (or g/cm³)
$m$	Mass	kg (or g)
$V$	Volume	m³ (or cm³)

Rearranging

To find mass: $m = \rho V$

To find volume: $V = \frac{m}{\rho}$

Unit conversion

$1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$

Water has a density of $1.0\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$.

Worked example

A metal block has a mass of 540 g and dimensions 10 cm × 6 cm × 2 cm. Calculate the density. [3]

Step 1. Calculate volume: $V = 10 \times 6 \times 2 = 120\ \text{cm}^3$

Step 2. Calculate density: $\rho = \frac{540}{120} = 4.5\ \text{g/cm}^3$

(In SI: $\rho = 4500\ \text{kg/m}^3$)

Marking points: M1 for calculating volume. M1 for $\rho = m/V$ with correct substitution. A1 for 4.5 g/cm³ or 4500 kg/m³.

These marking points are for this original example and are not an official Cambridge mark scheme.

Common errors and how to correct them

• Mixing units. If mass is in grams and volume in cm³, the density is in g/cm³. If mass is in kg and volume in m³, the density is in kg/m³. Do not mix (g with m³ or kg with cm³) without converting.
• Using diameter instead of radius for cylinders. Volume of a cylinder is $\pi r^2 h$. Use the radius (half the diameter), not the diameter.
• Forgetting to convert cm³ to m³. There are $10^6$ cm³ in 1 m³ (not 100). Multiply cm³ by $10^{-6}$ to get m³.

How to apply this in an exam

Show the volume calculation as a separate step before the density calculation. This earns an independent mark. Always state the unit alongside the final answer.