Volume and Surface Area

In Grade 6 you found the volume of a cuboid with $\text{length} \times \text{width} \times \text{height}$. In Stage 8 you discover that this is just one case of a much bigger idea — the prism.

A prism is a 3D solid that has the same shape all the way through. If you sliced it anywhere along its length, every slice would be identical. That repeated shape is called the cross-section.

A cuboid is a prism with a rectangular cross-section. A triangular prism has a triangular cross-section. There are also hexagonal prisms, and many more.

Why does this matter? Because instead of learning a separate formula for every solid, you learn one rule for all prisms. That is the real power of the cross-section idea — and it makes Stage 8 work feel much simpler than it first looks.

Here is the single most useful formula in this topic: $\text{Volume} = \text{cross-section area} \times \text{length}.$

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

For the triangular prism above:

• The cross-section is a triangle with base $6\text{ cm}$ and height $4\text{ cm}$. Its area is $\tfrac{1}{2} \times 6 \times 4 = 12\text{ cm}^2$.
• The length of the prism is $10\text{ cm}$.
• Volume $= 12 \times 10 = 120\text{ cm}^3$.

Notice the method works for any prism — the only part that changes is which area formula you use for the cross-section. A rectangular cross-section uses $l \times w$; a triangular one uses $\tfrac{1}{2}bh$; a trapezium cross-section uses the trapezium formula.

Two reminders: keep all measurements in the same unit, and write the answer in cubic units.

A net is the flat shape you get by unfolding a solid along its edges. Folding the net back up gives the solid again.

A triangular prism unfolds into a net with 5 faces: 2 triangles (the cross-sections) and 3 rectangles (the long sides).

Nets are useful in two ways. First, they help you see a solid clearly — every face is laid flat in front of you. Second, they make finding surface area straightforward, because the total area of the net is exactly the surface area of the solid.

A good check for any net: it must fold up with no overlaps and no gaps. When you draw a net, label each face — "triangle", "rectangle" — so you know you have included them all when you add up the areas.

Different prisms give different nets, but the pattern is always the same: a strip of rectangles for the sides, plus two copies of the cross-section.

The surface area of a solid is the total area of all its faces — picture the wrapping paper needed to cover it exactly. It is an area, so it is measured in square units.

The reliable method is: sketch the net, find the area of every face, then add them all.

For this triangular prism — cross-section base $6$, height $4$, sloping sides $5$ and $5$, prism length $10$:

• Two triangles: each is $\tfrac{1}{2} \times 6 \times 4 = 12$, so together $2 \times 12 = 24\text{ cm}^2$.
• Rectangle (bottom): $6 \times 10 = 60\text{ cm}^2$.
• Two sloping rectangles: each is $5 \times 10 = 50$, so together $2 \times 50 = 100\text{ cm}^2$.

Total surface area $= 24 + 60 + 100 = 184\text{ cm}^2$.

The most common slip is missing a face. Drawing the net first, and ticking off each face as you add it, makes that almost impossible.

Volume and surface area are not just formulae — they answer real questions about the world.

Volume tells you how much a container holds. There is a neat link to capacity: $1\text{ cm}^3$ holds exactly $1\text{ ml}$ of liquid, and $1000\text{ cm}^3$ holds $1$ litre. So a prism-shaped tank with volume $5000\text{ cm}^3$ holds $5$ litres.

Surface area tells you how much material covers a solid — wrapping paper for a gift, paint for a wall, or metal for a box.

For the tank above: volume $= 50 \times 30 \times 40 = 60\,000\text{ cm}^3$, which is $60$ litres of water.

A clear plan for any worded problem:

• Decide whether you need volume (space inside, how much it holds) 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 prism work you have done this year is the foundation for plenty ahead:

• Cylinders — a cylinder is like a prism with a circular cross-section, so its volume is circle area × length.
• Compound solids — bigger shapes are often prisms joined together, so you split, find each volume and add.
• Capacity and density — volume connects to how much a container holds and how heavy a material is.
• In everyday life — packaging, construction, plumbing and design all rely on volume and surface area.

If a later 3D topic feels tricky, the prism idea — cross-section times length — is almost always underneath it. Keep this guide handy.