Leave lesson

Geometry & measures · Perimeter, area & volume

1 / 9

Volume of prisms, cylinders, cones & spheres

Why a prism's volume is cross-section × length, why a cylinder is just a circular prism, where the ⅓ in pyramid and cone formulas fits in, spheres and hemispheres, and litres.

Geometry & measures · Perimeter, area & volume

Volume of prisms, cylinders, cones & spheres

Why a prism's volume is cross-section × length, why a cylinder is just a circular prism, where the ⅓ in pyramid and cone formulas fits in, spheres and hemispheres, and litres.

Why it works

Volume counts unit cubes. A cuboid is l×w×hl \times w \times h layers of them. The powerful idea behind every prism is slicing: a prism is any solid with the same cross-section all the way through, so it is a stack of identical slices —

V=area of cross-section×length.V = \text{area of cross-section} \times \text{length}.

Find the area of the end face first (a triangle, a trapezium, an L-shape — whatever it is), then multiply by how far it runs. Multiplying together every number printed on the diagram is not a method.

A cylinder is a circular prism. Cross-section πr2\pi r^2, so V=πr2hV = \pi r^2 h. The one thing to check before squaring: is the printed measurement the radius or the diameter? Halve the diameter first — squaring a diameter quadruples the answer.

Pointy solids take a third. A pyramid fills exactly 13\frac{1}{3} of the prism with the same base and height, and a cone is a circular pyramid:

Vpyramid=13×base area×h,Vcone=13πr2h.V_{\text{pyramid}} = \tfrac{1}{3} \times \text{base area} \times h, \qquad V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h.

Sphere: V=43πr3V = \frac{4}{3}\pi r^3. Edexcel prints the sphere and cone formulas in the question when you need them — your job is to substitute correctly, keep the 43\frac{4}{3} or 13\frac{1}{3}, and cube (not square) the radius. A hemisphere is half the sphere's volume: 23πr3\frac{2}{3}\pi r^3.

π answers: exact or rounded. On the non-calculator paper, leave answers "in terms of π\pi" — 48π48\pi cm³ is exact and expected. On calculator papers, keep the full display value and round only at the end.

Capacity: 11 litre =1000= 1000 cm³, and 11 m³ =1000= 1000 litres. Water questions usually want a volume converted into litres — or a "how many times does the jug fill the tank" division.