Volume of a Rectangular Prism: Length × Width × Height

Counting cubes

Volume is how much space is inside a solid, counted in unit cubes — a cube one unit long in every direction.

$$V = l \times w \times h$$

A box $6$ long, $4$ wide and $3$ tall:

$$V = 6 \times 4 \times 3 = 72 \text{ cubic units}$$

Why multiplying three lengths counts cubes

The calculation is the counting, done in two stages.

First, cover the floor. The base is a rectangle $6$ by $4$, so it holds $6 \times 4 = 24$ cubes. That is exactly the area of a rectangle, and this is where it earns its keep.

Then stack. The box is $3$ cubes tall, so there are $3$ identical layers of $24$:

$$24 \times 3 = 72$$

Multiplication counts equal groups, and the layers are equal groups.

Why the units are cubed

Each unit cube is one centimetre wide, one tall and one deep — one cubic centimetre. Counting $72$ of them gives $72$ cm³.

Quantity	Lengths multiplied	Units
perimeter	one	cm
area	two	cm²
volume	three	cm³

The exponent counts the dimensions. This is the fastest check on any answer: if three lengths were multiplied, the units carry a $3$, and if they do not, something was missed.

Base times height

The same formula is often written

$$V = B h$$

where $B$ is the area of the base. It is not a different rule — $B$ is already $l \times w$, so this is the same product grouped differently.

The advantage is that it keeps working when the base is not a rectangle. A prism with a triangular base has volume $B h$ too, with $B$ being the triangle’s area. One formula, any prism.

Volume and surface area are different

	Measures	Units
volume	what fits inside	cm³
surface area	the wrapping outside	cm²

A box that holds two litres and a box that takes two square metres of paper are answering unrelated questions. The units keep them apart.

Worked examples

Common mistakes

Practice problems