Surface Area and Volume of Prisms

A rectangular prism (a box) has length $l$, breadth $b$ and height $h$.

• Volume — the space inside — is $V = l \times b \times h$ (measured in cubic units, $\text{cm}^3$).
• Surface area — the total area of all six faces — is $2(lb + lh + bh)$ (measured in square units, $\text{cm}^2$).

A cube is the special case where all edges are equal ($s$): volume $= s^3$ and surface area $= 6s^2$.

Worked example. A box $4 \times 3 \times 2$: volume $= 4 \times 3 \times 2 = 24$; surface area $= 2(4{\cdot}3 + 4{\cdot}2 + 3{\cdot}2) = 2(12 + 8 + 6) = 52$.

Exam checkpoint

Surface area measures the outside covering of a 3D object; volume measures the space it holds. A rectangular prism has three pairs of matching faces. List each face area once, double the total, and use square units. For volume multiply length, width and height and use cubic units.

Short worked example. A $4\text{ cm}\times3\text{ cm}\times2\text{ cm}$ box has volume $4\times3\times2=24\text{ cm}^3$. Its surface area is $2(12+8+6)=52\text{ cm}^2$.