Edexcel · GCSE Maths · 1MA1 · Foundation and Higher
M31 · Solids, surface area and volume
Revision notes, worked examples and methods for solids, surface area and volume.
Notes and quizzes ready · 50 questions · Video coming soon.
Foundation shows shared notes and questions. Higher includes the shared content and labelled Higher extensions. Changing tier starts the notes and a new quiz set.
Revise the key ideas
Solids, views and units
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
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.
Volumes
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
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 areas
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
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.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
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