Area and volume of common shapes

What the exam asks

Expect the formulae to arrive in the stem and the reasoning not to. The commonest items are scaling questions, surface area to volume comparisons dressed as biology, and volume calculations where the answer wanted is a ratio. The trap that catches most candidates is scaling linearly: told that a cell's diameter doubles, they double the volume, and every downstream conclusion about diffusion or demand is then wrong. Whenever a length changes, ask what power that length is raised to in the quantity you are asked about, then raise the factor to that power. The second trap is unit conversion: a cubic metre is a million cubic centimetres, not a hundred, because the factor of 100 between the lengths gets cubed along with everything else.

The formulae here are short and mostly supplied when they are needed. What is not supplied, and what the exam actually leans on, is the scaling rule: area goes as the square of a length and volume goes as the cube. Double every length of a shape and its area quadruples while its volume grows eightfold. That one sentence answers more questions in this topic than all the formulae together, and it works for any shape at all, not just the ones with names.

The consequence that matters most is the surface area to volume ratio. Because volume climbs faster than surface, a bigger object has less surface per unit of volume, and for a sphere the ratio is just 3 divided by the radius. This is the arithmetic under a large amount of Section III biology: why cells stay small, why an intestine is folded into villi, why alveoli are numerous rather than large, and why a small mammal loses heat faster than a big one. If a stem describes something whose function depends on exchange across a surface, the ratio is likely to be the point.

Most of these questions ask for a factor, not a value. When they do, write the ratio of the two cases and watch pi, the one third, and every other constant cancel. A question that looks like it needs a calculator usually needs a division you can do in your head, once you notice that nothing shared by both cases has to be computed.

What to hold

• Scale every length of any shape by a factor k and its area scales by k squared while its volume scales by k cubed, whatever the shape is.
• Doubling a length therefore quadruples area and multiplies volume by eight, and halving a length divides area by four and volume by eight.
• The surface area to volume ratio falls as an object grows, because volume rises with the cube while surface rises only with the square.
• For a sphere the surface area to volume ratio is 3 over the radius, and for a cube it is 6 over the side length, so both are inversely proportional to size.
• A circle has area pi r squared and circumference 2 pi r, so area depends on the square of the radius while the perimeter depends on it directly.
• A sphere has volume four thirds pi r cubed and surface area 4 pi r squared.
• Any prism or cylinder has a volume equal to its cross-sectional area times its length, which is one rule covering every uniform shape.
• A cone or a pyramid has one third the volume of the prism or cylinder that shares its base and height.
• The area of a triangle is half base times perpendicular height, and the height must be perpendicular to the base, not the length of a sloping side.
• One millilitre is one cubic centimetre and one litre is 1000 cubic centimetres, so one cubic metre is 1000 litres.
• Converting units of area squares the length factor and units of volume cubes it: one square metre is 10 to the 4 square centimetres, and one cubic metre is 10 to the 6 cubic centimetres.
• When a question asks for a factor rather than a value, take the ratio of the two cases and every constant common to both cancels, pi included.