Surface area and volume
Maths · GCSE · The School
Every prism is one idea
A prism has the same cross-section all the way through, and its volume is the area of that cross-section multiplied by the length. That single sentence covers cuboids, triangular prisms and cylinders, so there are far fewer formulas here than the topic first appears to have. For a cylinder the cross-section is a circle, so the volume is πr²h. Surface area is different in kind: it is not one formula but the total of the faces, so the reliable method is to list the faces, find each area, and add.
The cylinder, opened out
Cut a cylinder down its side and roll it flat and you get a rectangle whose width is the circumference and whose height is the height of the cylinder. So the curved surface is 2πr × h. Add the two circular ends, πr² each, and the total surface area is 2πrh + 2πr². Whether to include the ends depends on the object: a tin of beans has both, a pipe has neither, a bucket has one. Read the question for which of those it is rather than reaching for the full formula every time.
Cones and spheres
These four are given on the formula sheet but must be recognised. A cone has volume ⅓πr²h and curved surface area πrl, where l is the SLANT height, not the vertical height — and if the question gives you the vertical height you find the slant with Pythagoras, since l² = r² + h². A sphere has volume 4/3 πr³ and surface area 4πr². The most common error in the whole topic is putting the vertical height into πrl, which is why the slant is worth marking on the diagram before starting.
Area is squared and volume is cubed, and the conversions follow the same power: 1 cm² is 100 mm², and 1 cm³ is 1000 mm³. Enlarge a solid by scale factor k and areas grow by k² while volumes grow by k³ — doubling every length multiplies the volume by eight. A cylinder holds 500 cm³ and is 10 cm tall; find its radius. Start from V = πr²h: 500 = πr² × 10, so r² = 500 / (10π) = 15.9, and r = 3.99 cm. Working backwards is the same formula rearranged, and questions ask for it as often as they ask forwards.