Volume and Surface Area

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.

• Volume of a prism = cross-section area × length.
• The cross-section is one repeated 'layer' of the solid.
• Use the matching area formula for the cross-section shape.
• Volume answers are always in cubic units.

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.

• A cylinder is a prism with a circular cross-section.
• Volume of a cylinder = π × r² × h.
• Square the radius first, then multiply by π and the height.
• Given the diameter? Halve it to find r.

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 cylinder's net is 2 circles + 1 rectangle.
• The unrolled curved side has width 2πr and height h.
• Surface area = 2πr² + 2πrh.
• Surface area is always in square units.

• 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.

• A composite solid splits into prisms and cylinders.
• Find each volume separately, then add them up.
• For a hollow solid, subtract the empty part.
• Hidden joining faces are not part of the surface area.

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.

• Volume answers 'how much does it hold?'
• 1 cm³ holds 1 ml; 1000 cm³ holds 1 litre.
• Surface area answers 'how much material covers it?'
• Decide which one a problem needs before you start.