🧊Volume & Surface Area

Two different questions can be asked about any solid.

Volume asks how much space it fills — how many unit cubes fit inside. Three lengths are multiplied, so the units are cubic: cm³, m³.

Surface area asks how much material would cover the outside — how much wrapping paper, paint or card. It is a sum of flat faces, so the units are square: cm², m².

Keeping the units in view is the fastest way to spot a mistake. A volume answered in cm² is definitely wrong.

A useful mental image: volume is the water a tank holds; surface area is the metal the tank is made of.

A rectangular prism (a box) with length l, width w and height h holds

V = l x w x h

Think of it as layers: the bottom layer holds l x w cubes, and there are h such layers stacked up.

Its surface is six rectangles in three matching pairs, so

S = 2(lw + lh + wh)

A cube is the special case where all three edges are equal:

V = s³ S = 6 s²

The 6 x (one face) shortcut works only for a cube. In a general box the three pairs of faces are different sizes, so all three products are needed.

A cylinder is a stack of identical circular discs. Its volume is therefore the area of the circular base times the height:

V = pi x r² x h

For radius 3 and height 10 that is pi x 9 x 10 = 270*pi, or about 847.8 using pi = 3.14.

As with circles, the question will say whether it wants an exact answer in terms of pi or a decimal using pi = 3.14.

The commonest slips are doubling the radius instead of squaring it, and stopping after the base area without multiplying by the height.

The curved surface of a cylinder unrolls into a rectangle of width 2 pi r (the circumference) and height h, so the total surface area of a closed cylinder is 2 pi r² + 2 pi r h.

Three solids taper to a point or curve away, and each has its own factor in front.

A cone fills exactly one third of the cylinder standing on the same base circle with the same height:

V = (1/3) x pi x r² x h

A pyramid does the same to its prism — three identical pyramids fill the prism on the same base — so with B for the area of the base,

V = (1/3) x B x h

For a square base of edge b that means B = b x b, and the volume is b x b x h / 3.

A sphere has no base to stand on, and its factor is four thirds:

V = (4/3) x pi x r³

The radius is cubed here, not squared. Half a sphere — a hemisphere — is half of that, which tidies to (2/3) x pi x r³.

Two warnings that catch nearly everyone. The h in a cone is the perpendicular height, measured straight up from the centre of the base to the tip; the slant height runs up the sloping side and belongs to surface area, never to volume. And a question that hands you a diameter is asking you to halve it first — using it as the radius makes a cone or a sphere far too big.

Composite solids are just addition. A tank made of a cylinder with a hemispherical dome holds pi x r² x h plus (2/3) x pi x r³: find each piece, then add.

Because volume is built by multiplying, it is undone by dividing. If a tank has volume 480 cm³ and a base measuring 8 cm by 6 cm, then the base area is 48 cm² and the depth is 480 / 48 = 10 cm.

Capacity is volume wearing different units. The key conversion is

1 litre = 1000 cm³

so a volume in cm³ is divided by 1000 to reach litres. Converting to a bigger unit always makes the number smaller, which is a good check: 4500 cm³ is 4.5 litres, not 4 500 000.

A related trap is scale. Doubling every edge of a box multiplies its surface area by 4 and its volume by 8 — this is why large animals are shaped so differently from small ones.