Calculate the volume and surface area of cuboids, prisms and cylinders, and solve problems involving capacity

What this dot point is asking

SEAB wants you to find the volume and surface area of cuboids, prisms and cylinders, and to handle capacity problems (such as how much liquid a container holds). Volume measures the space inside a solid; surface area measures the total area of its outer faces. Both rely on the plane-figure area formulae applied carefully.

The answer

Volume

Volume is the amount of space a solid occupies, measured in cubic units (cm$^3$, m$^3$). The main formulae are:

• Cuboid: $\text{volume} = \text{length} \times \text{width} \times \text{height}$.
• Prism: $\text{volume} = \text{area of cross-section} \times \text{length}$.
• Cylinder: $\text{volume} = \pi r^2 h$ (a prism with a circular cross-section).

A prism has the same cross-section all the way along, so its volume is always the cross-sectional area times the length.

Surface area

Surface area is the total area of all the outer faces, measured in square units. To find it, work out the area of each face and add them up.

• A cuboid has three pairs of equal rectangular faces.
• A cylinder has two circular ends (each $\pi r^2$) and a curved surface that unrolls into a rectangle of area $2\pi r h$, giving total surface area $2\pi r^2 + 2\pi r h$.

Capacity

Capacity is the volume a container can hold, usually given in litres or millilitres for liquids. A useful conversion is:

So a tank with a volume of $5000\ \text{cm}^3$ holds $5$ litres.

Keeping units consistent

Work in a single unit throughout. If some lengths are in metres and others in centimetres, convert them all to the same unit before calculating, or the volume will be wrong by a large factor.

Examples in context

Example 1. A swimming pool. A rectangular pool $25\ \text{m}$ long, $10\ \text{m}$ wide and $2\ \text{m}$ deep holds a volume of $25 \times 10 \times 2 = 500\ \text{m}^3$ of water. Treating the pool as a cuboid turns a real container into a simple volume calculation, useful for working out how much water it takes to fill.

Example 2. A triangular prism tent. A tent shaped like a triangular prism has a triangular cross-section and a length along the ground. Its volume is the triangle's area times the tent's length. Recognising the prism shape means you only need the cross-section area and the length, no matter how unusual the solid looks.

Try this

• Cue. Find the volume of a cuboid $8\ \text{cm}$ by $5\ \text{cm}$ by $2\ \text{cm}$. Compute $8 \times 5 \times 2 = 80\ \text{cm}^3$.
• Cue. A cylinder has radius $2\ \text{cm}$ and height $7\ \text{cm}$. With $\pi = \dfrac{22}{7}$, volume $= \dfrac{22}{7} \times 4 \times 7 = 88\ \text{cm}^3$.
• Cue. Convert $3500\ \text{cm}^3$ to litres. Divide by $1000$ to get $3.5$ litres.