Volume and Surface Area

A 3D solid is a shape that takes up space — it has length, width and height. A ball, a box and an ice-cream cone are all 3D.

We describe a solid using three parts:

• A face is a flat surface of the solid.
• An edge is a line where two faces meet.
• A vertex is a corner where edges meet (the plural is vertices).

Some solids to recognise: a cube (6 square faces), a cuboid (6 rectangular faces), a cylinder (2 circular faces and one curved surface), a sphere (one curved surface), a cone and a pyramid.

A cuboid has 6 faces, 12 edges and 8 vertices. Counting these is a neat way to tell solids apart.

Imagine carefully cutting along the edges of a cardboard box and flattening it out. The flat shape you are left with is called a net.

A net of a cube is made of 6 squares arranged so they fold up without overlapping. One common arrangement looks like a cross:

A net of a cuboid uses 6 rectangles instead — in 3 matching pairs, because opposite faces of a cuboid are the same size.

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

A good check: a correct net must fold up with no faces overlapping and no gaps left open. Not every arrangement of 6 squares works — try sketching a few and folding them in your head.

Volume is the amount of space a 3D solid takes up — how much it would hold if you filled it. It is measured in cubic units such as cubic centimetres ($\text{cm}^3$) or cubic metres ($\text{m}^3$).

A cubic centimetre is the space taken up by a small cube measuring 1 cm along every edge. To find the volume of a solid, you are really counting how many of these unit cubes fit inside.

Notice the small raised 3 in $\text{cm}^3$. It is there because volume uses three measurements — length, width and height — all multiplied together. (Area uses two, so it has a small 2: $\text{cm}^2$.)

Volume is a useful idea in real life. It tells you how much water a tank holds, how much soil fills a plant pot, or how much space a parcel takes up in a delivery van.

A cuboid is a box shape with 6 rectangular faces. Finding its volume is wonderfully simple:

$\text{Volume} = \text{length} \times \text{width} \times \text{height}.$

For the cuboid above, with length 6 cm, width 3 cm and height 4 cm:

$\text{Volume} = 6 \times 3 \times 4 = 72\text{ cm}^3.$

Two things to be careful about:

• Same units. All three measurements must use the same unit. If one is in metres and another in centimetres, convert first.
• Cubic units. Because you multiplied three lengths, the answer is in cubic units — $\text{cm}^3$ here.

A cube is just a special cuboid where length, width and height are all equal. So the volume of a cube with edge 5 cm is $5 \times 5 \times 5 = 125\text{ cm}^3$.

You can also work backwards: if you know the volume and two measurements, divide to find the third.

The surface area of a solid is the total area of all its faces — imagine the amount of wrapping paper needed to cover it exactly. It is an area, so it is measured in square units like $\text{cm}^2$.

A cuboid has 6 rectangular faces, and opposite faces are identical. So there are 3 matching pairs:

The neat method is: find the area of three different faces, add them, then double.

For a cuboid with length 6 cm, width 3 cm and height 4 cm:

• Top (and bottom): $6 \times 3 = 18\text{ cm}^2$.
• Front (and back): $6 \times 4 = 24\text{ cm}^2$.
• End (and end): $3 \times 4 = 12\text{ cm}^2$.

Add the three: $18 + 24 + 12 = 54\text{ cm}^2$. Then double for the matching pairs: $54 \times 2 = 108\text{ cm}^2$.

So the surface area is $108\text{ cm}^2$. Drawing the net first makes it easy to be sure you have counted all 6 faces.

Getting confident with cuboids sets you up for plenty of later geometry:

• Prisms and cylinders — their volumes build on the same idea of an area multiplied by a length.
• Capacity — volume links directly to how much liquid a container holds ($1\text{ cm}^3$ holds $1\text{ ml}$).
• Compound solids — bigger shapes are often just cuboids joined together, so you split and add.
• In everyday life — packing, shipping, building, painting and even cooking all use volume and surface area.

If a later 3D topic feels tricky, it is often the cuboid skills underneath that need a quick refresh. Come back to this guide whenever you need it.