Area, volume and surface area
The standard formulae, and choosing the right one.
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 5.1.1Calculate perimeter and area of common shapes including circles and compound shapes.
- 5.1.2Calculate surface area and volume of prisms, cylinders, cones and spheres.
6 minute read
Area, volume and surface area
Areas worth knowing cold
- Rectangle:
length × width - Triangle:
½ × base × height(perpendicular height, not the slant side) - Parallelogram:
base × perpendicular height - Trapezium:
½(a + b) × h - Circle:
A = πr², circumferenceC = 2πrorπd
Volumes
- Prism (constant cross-section):
area of cross-section × length - Cylinder:
πr²h - Cone:
⅓πr²h - Sphere:
4/3 πr³ - Pyramid:
⅓ × base area × height
The cone and pyramid both carry the ⅓, which is the factor most often dropped.
Surface areas
- Cylinder (closed):
2πr² + 2πrh— two circles plus the curved surface, which unrolls into a rectangle of width2πrand height h. - Cone:
πr² + πrl, where l is the slant height, not the vertical height. - Sphere:
4πr²
If a question says "open cylinder" or "no lid", leave out one or both circles — read carefully.
Compound shapes
Split into standard shapes, calculate each, then add or subtract. Label each part so the method is visible to the examiner.
Units — a reliable source of lost marks
Area is in square units, volume in cubic units. When converting, remember 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³, because the conversion factor is squared or cubed.
Think of it like this
The curved surface of a cylinder is a label peeled off a tin: it unrolls into a rectangle whose width is the circumference. That is exactly where `2πrh` comes from, so the formula need not be memorised blindly.
Worked examples
Method, step by step
A cylinder has radius 5 cm and height 12 cm. Find its volume and total surface area, to 3 significant figures.
- 1Volume = πr²h = π × 5² × 12 = 300π ≈ 942.48 cm³.
- 2Surface area = 2πr² + 2πrh.
- 32π(5²) = 50π and 2π(5)(12) = 120π.
- 4Total = 170π ≈ 534.07 cm².
Volume ≈ 942 cm³; surface area ≈ 534 cm²
Common misconceptions
- Using the slant height in the cone **volume** formula. Volume uses the perpendicular height h; the slant height l belongs in the curved surface area.
- Converting areas with a linear factor — 1 m² is 10 000 cm², not 100.
- Forgetting the ⅓ in cone and pyramid volumes, which triples the answer.
In the exam
- Write the formula down before substituting. It earns method marks even if the arithmetic later goes wrong.
- Keep π in your calculator to full precision and round only at the end, unless asked to leave the answer in terms of π.