Mr Kidman's Classroom
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 prism’s cross-section is the shape of a slice taken at right angles to its length; it stays the same all along. An apex is the vertex where the pyramid’s triangular faces meet.
- 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 ( × 4 × 3) × 8 = 48 cm3.
- A pyramid has V = × base area × perpendicular height. A cone has V = πr2h. 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 = πr3. A hemisphere has half this volume: πr3.
- Worked example: A cone with radius 3 cm and perpendicular height 8 cm has volume × π × 9 × 8 = 24π cm3.
Surface areas
- Surface area of a cuboid is 2lw + 2lh + 2wh. A net (a flat layout of the unfolded faces) 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 surface that makes a rectangle when unwrapped. An open-top cylinder has only one circular end.
That rectangle has width 2πr (the circumference) and height h, giving area 2πrh.
- 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.
The two ends give 2 × π × 22 = 8π cm2; the curved side gives 2π × 2 × 5 = 20π cm2. Volume instead uses π × 22 × 5 = 20π cm3.
- 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.
