Volume and Surface Area

In Grade 7 you discovered the prism — a solid with the same cross-section all the way through. In Stage 9 you build on that with one reliable rule: $\text{Volume} = \text{cross-section area} \times \text{length}.$

The cross-section is one "layer", and the length tells you how many layers thick the solid is.

For the triangular prism above — cross-section a triangle of base $6\text{ cm}$ and height $4\text{ cm}$, prism length $12\text{ cm}$:

• Cross-section area $= \tfrac{1}{2} \times 6 \times 4 = 12\text{ cm}^2$.
• Volume $= 12 \times 12 = 144\text{ cm}^3$.

The method works for any prism — only the area formula for the cross-section changes. That single idea is about to unlock the cylinder, because a cylinder is simply a prism whose cross-section is a circle.

Two reminders that always apply: keep every measurement in the same unit, and write a volume answer in cubic units.

A cylinder is a solid with two identical circular ends and a curved side — a tin can is a perfect example. Because its cross-section is a circle that never changes, a cylinder is just a prism.

So the prism rule applies. The cross-section area is the area of a circle, $\pi r^2$, and the "length" is the height $h$: $\text{Volume} = \pi \times r^2 \times h.$

For the cylinder above — radius $4\text{ cm}$, height $10\text{ cm}$:

• Cross-section (circle) area $= \pi \times 4^2 = \pi \times 16 \approx 50.3\text{ cm}^2$.
• Volume $= 50.3 \times 10 \approx 503\text{ cm}^3$.

The order still matters: square the radius first, then multiply by π, then multiply by the height. And if a problem gives you the diameter, halve it to find $r$ before you start.

The surface area of a cylinder is the total area of every part of its outside. The clearest way to see it is to unroll the cylinder into its net.

The net has three pieces: two circles (the top and bottom) and one rectangle (the curved side, unrolled flat).

Here is the clever part. When you unroll the curved side, it becomes a rectangle. Its height is the cylinder's height $h$, and its width is the circumference of the circle, $2\pi r$ — because that is the distance once around the can.

So the total surface area is: $\text{Surface area} = \underbrace{2 \times \pi r^2}_{\text{two circles}} + \underbrace{2\pi r \times h}_{\text{curved side}}.$

For a cylinder with radius $4\text{ cm}$ and height $10\text{ cm}$:

• Two circles: $2 \times \pi \times 4^2 \approx 100.5\text{ cm}^2$.
• Curved side: $2 \times \pi \times 4 \times 10 \approx 251.3\text{ cm}^2$.
• Total $\approx 351.8\text{ cm}^2$.

The most common slip is forgetting a circle, or leaving out the curved side altogether. The net is your checklist — three pieces, every time.

A composite solid is built from simpler solids joined together — a cuboid with a cylinder on top, or two prisms stuck end to end.

The method mirrors compound shapes in 2D: split the solid into prisms and cylinders, find each volume, then add them (or subtract a hollow part).

For example, a cuboid base $10\text{ cm} \times 6\text{ cm} \times 4\text{ cm}$ with a cylinder of radius $3\text{ cm}$ and height $7\text{ cm}$ on top:

• Cuboid: $10 \times 6 \times 4 = 240\text{ cm}^3$.
• Cylinder: $\pi \times 3^2 \times 7 \approx 197.9\text{ cm}^3$.
• Total volume $\approx 240 + 197.9 = 437.9\text{ cm}^3$.

A reliable plan:

• Identify each simple solid and label its measurements.
• Find every missing length by adding or subtracting known lengths.
• Add the volumes — or, for a hollow solid, subtract the empty part.
• For surface area, be careful: where two solids join, those touching faces are hidden, so do not include them.

Volume and surface area answer real questions about containers, materials and design.

Volume tells you how much a container holds. There is a neat link to capacity: $1\text{ cm}^3$ holds exactly $1\text{ ml}$, and $1000\text{ cm}^3$ holds $1$ litre.

For the tank above — radius $20\text{ cm}$, height $50\text{ cm}$: $\text{Volume} = \pi \times 20^2 \times 50 \approx 62\,800\text{ cm}^3.$ That is about $62\,800\text{ ml}$, or roughly $62.8$ litres of water.

Surface area tells you how much material covers a solid — metal for a tin, paint for a tank, or labels for a can.

A clear plan for any worded problem:

• Decide whether you need volume (space inside) or surface area (covering material).
• Check every measurement is in the same unit first.
• Choose the right formula, work carefully, and label the units in your answer.

The volume and surface area work you have done in Stage 9 is the foundation for plenty ahead:

• Cones, pyramids and spheres — solids that taper or curve, with their own volume formulae built on these ideas.
• Density — mass divided by volume; you will combine volume with mass to compare materials.
• Similar solids — when a solid is scaled up, its volume and surface area change in predictable ways.
• In everyday life — packaging design, plumbing, construction and manufacturing all rely on volume and surface area.

If a later 3D topic feels tricky, the cross-section idea — and the cylinder's net — is almost always underneath it. Keep this guide close.