M31 · Solids, surface area and volume | Edexcel GCSE Maths

Revise the key ideas

A face is a surface, an edge is where faces meet, and a vertex is a corner. A cuboid has 6 faces, 12 edges and 8 vertices. A sphere has one curved surface and no edges or vertices.
A prism has a constant cross-section along its length. A pyramid has a base and triangular faces meeting at an apex. Cylinders and cones have curved surfaces, so not every face is flat.
A plan is the view from above; an elevation is a specified side or front view. Draw the visible outline using the given dimensions, not a perspective sketch.
Cuboid plan and front elevation
Enlarge diagram: Cuboid plan and front elevation
Volume measures space occupied and uses cubic units; surface area totals the external surfaces and uses square units. Capacity may use litres, with 1 litre = 1000 cm³.

Volume of a cuboid = lwh. Volume of a prism = cross-sectional area × perpendicular length. A cylinder has V = πr²h.
Worked example: A triangular prism with triangle base 4 cm, perpendicular height 3 cm and prism length 8 cm has volume (12 × 4 × 3) × 8 = 48 cm³.
A pyramid has V = 13 × base area × perpendicular height. A cone has V = 13πr²h. Slant height is not the height in these volume formulae.
In a right circular cone, perpendicular height h, radius r and slant height l form a right triangle: l² = h² + r². Use h for volume and l for curved surface area.
Perpendicular height and slant height in a cone
Enlarge diagram: Perpendicular height and slant height in a cone
A sphere has V = 43πr³. A hemisphere has half this volume: 23πr³.
Worked example: A cone with radius 3 cm and perpendicular height 8 cm has volume 13 × π × 9 × 8 = 24π cm³.

Surface area of a cuboid is 2lw + 2lh + 2wh. A net helps identify every face and avoid counting a hidden internal join.
A net with all six cuboid faces
Enlarge diagram: A net with all six cuboid faces
For a closed cylinder, total surface area is 2πr² + 2πrh: two circular ends and one curved rectangular surface. An open-top cylinder has only one circular end.
A cone's curved surface area is πrl, where l is slant height. Add πr² for its circular base when finding total surface area.
A sphere's surface area is 4πr². A hemisphere has curved area 2πr², or total area 3πr² if its circular base is included.
Worked example: A closed cylinder of radius 2 cm and height 5 cm has volume 20π cm³ and total surface area 8π + 20π = 28π cm².
For composite volumes, add non-overlapping parts or subtract a removed region. For composite surface areas, omit faces hidden where parts join. State which surfaces the question requires.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.