lesson
Describe a solid, including what is hidden
1 · Learn
A polyhedron has flat polygonal faces. An edge is where two faces meet, and a vertex is where edges meet. A cube has 6 square faces, 12 edges and 8 vertices. A cuboid has 6 rectangular faces, 12 edges and 8 vertices; a cube is a special cuboid. A sketch hides some features, so count the solid, not just the visible lines.
A prism has two matching parallel polygonal ends with corresponding vertices joined. A triangular prism has 2 triangular ends and 3 rectangular side faces when it is a right prism: 5 faces, 9 edges and 6 vertices. In an n-sided prism there are n+2 faces, 3n edges and 2n vertices. The diagram makes all nine edges visible or dashed.
A pyramid has one polygonal base and triangular faces meeting at an apex. For an n-sided base, there are n+1 faces, 2n edges and n+1 vertices. A square-based pyramid therefore has 5 faces, 8 edges and 5 vertices. It is not a prism: its cross-sections parallel to the base shrink towards the apex.
Curved solids need different descriptions. A cylinder has two flat circular ends, one curved surface, two circular rims and no vertices. A cone has one flat circular base, one curved surface, one circular rim and one apex. A sphere has one curved surface, no flat faces, no edges and no vertices. Circular rims are not straight polyhedron edges. State this convention rather than mixing flat-face counts with curved surfaces.
2 · Worked example
A pentagonal prism has 5+2=7 faces, 3×5=15 edges and 2×5=10 vertices. A pentagonal pyramid has 5+1=6 faces, 2×5=10 edges and 5+1=6 vertices. Matching base names do not make them the same solid.
3 · Your turn
Count faces, edges and vertices of a hexagonal prism and a hexagonal pyramid. How does a cylinder differ from either polyhedron?
Check your answer
Prism: 8 faces, 18 edges, 12 vertices. Pyramid: 7 faces, 12 edges, 7 vertices. A cylinder has circular ends and a curved surface, not polygonal faces joined by straight edges.
4 · Apply your learning
Compare a safe classroom cube, cuboid, prism and pyramid. Mark each counted edge on a drawing, using dashed lines for hidden edges. Sort the solids by flat faces, curved surfaces and presence of an apex, giving a reason for each choice.
Adult guidance and safety
Use labelled diagrams and equal-scale coordinate grids. Check the stated properties rather than judging a sketch by appearance. Model transformations with tracing paper; use safe classroom solids. An adult should supervise any cutting or compass work.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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