Printable · GCSE Foundation · ages 14-16
Properties of 3D shapes worksheet — GCSE Foundation
Fifteen questions on "properties of 3d shapes" — DfE statement G12. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Answer key: Properties of 3D shapes worksheet — GCSE Foundation
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- (d) Cylinder — Method: check how many faces are flat and how many are curved, and how many flat faces there are. Working: this solid has a curved surface joining two flat circular faces of equal size, which is exactly a cylinder. A student who answers cone has forgotten a cone has only one flat circular face, not two. A student who answers sphere has forgotten a sphere has no flat faces at all. A student who answers triangular prism has spotted that the solid has two identical end faces but has taken those ends to be triangles rather than circles, and a prism's side faces are flat rectangles, not one curved surface. Answer: cylinder.
- (a) Square-based pyramid — A solid with one square base and four triangular faces meeting at a single apex above the base is a square-based pyramid. A triangular prism has two triangular faces and three rectangular faces, not a square base with four triangles, so that is a different solid. A cube has six square faces, and a cuboid has six rectangular faces — neither has any triangular faces at all. The solid described is a square-based pyramid.
- (a) 8 — A cylinder has two flat circular faces (the top and the base) and one curved surface, so only its 2 flat faces are wrapped. A cuboid has 6 faces and all of them are flat, so all 6 are wrapped. Adding these: 2 + 6 = 8, so 8 is correct. 9 comes from wrongly counting the cylinder's curved surface as a flat face. 6 comes from counting only the cuboid and forgetting the cylinder's two flat circular faces. 3 comes from counting only the cylinder and including its curved surface in that total.
- (a) 9 — A cube has 9 planes of symmetry in total: 3 that pass through the middles of pairs of opposite faces, and 6 more that pass through pairs of opposite edges diagonally. A candidate who counts only the 3 face-to-face planes — which is correct for a cuboid with three different edge lengths, but forgets that a cube's equal edges create 6 more diagonal planes — answers 3. A candidate who counts only the 6 diagonal planes and forgets the 3 face-to-face ones answers 6. A candidate who confuses the number of planes of symmetry with the number of edges on a cube answers 12. The correct total for a cube is 9.
- (a) 9 — A triangular prism has two triangular ends and three rectangular side faces. Each triangular end contributes 3 edges, giving 6 edges for both ends, and three more edges run lengthways to join the two ends together: 6 + 3 = 9 edges. Choosing 6 counts only the vertices (3 on each triangular end). Choosing 5 counts the faces (2 triangular + 3 rectangular) instead of the edges. Choosing 12 is the edge count of a cuboid, not a triangular prism.
- (d) 4 — Method: a plane of symmetry must pass through the apex and cut the base along one of the base's own lines of symmetry. Working: a square has 4 lines of symmetry (2 through opposite edge midpoints, 2 through opposite corners), and each of these, combined with the apex, gives one plane of symmetry of the pyramid. A student who answers 2 has only found the planes through the edge midpoints, or only the ones through the corners, and missed the other pair. A student who answers 8 has doubled the correct count, perhaps confusing it with a different solid. A student who answers 1 has only spotted the one obvious front-to-back plane. Answer: 4.
- (a) 5 — Method: add the two triangular end faces to the three rectangular side faces. Working: 2 triangular faces + 3 rectangular faces = 5 faces in total. A student who answers 6 has confused this with a cuboid's face count. A student who answers 9 has counted the edges instead of the faces. A student who answers 3 has only counted the rectangular faces and forgotten the two triangular ends. Answer: 5 faces.
- (c) 1 — Method: the instruction states that a curved surface counts as a face, so count the surfaces of the sphere on that basis. Working: a sphere has exactly one continuous curved surface and no flat surfaces at all. A student who answers 0 has ignored the instruction and refused to count the curved surface. A student who answers 2 has confused the sphere with a cylinder, which has two flat circular faces. A student who answers 3 has imagined extra hidden surfaces that do not exist. Answer: 1 face.
- (a) Triangular-based pyramid (tetrahedron) — A solid with 4 triangular faces, 4 vertices and 6 edges is a triangular-based pyramid, also called a tetrahedron. A triangular prism also has triangular faces, but it has 2 triangular faces plus 3 rectangular faces, 6 vertices and 9 edges — the extra rectangular faces and edges rule it out here. A square-based pyramid has 5 faces (one square, four triangles), 5 vertices and 8 edges, which does not match. A cube has 6 faces, 8 vertices and 12 edges, all much higher than the numbers given. The solid described is a triangular-based pyramid.
- (a) Triangular prism — Count the given properties against each solid. A triangular prism has 5 faces (2 triangular ends + 3 rectangular sides), 9 edges and 6 vertices — this matches exactly, so the solid is a triangular prism. A square-based pyramid also has 5 faces, but 8 edges and 5 vertices, and only one face is a square rather than three rectangles — the edge and vertex counts don't match. A cuboid has 6 faces, 12 edges and 8 vertices, none of which match the numbers given. A tetrahedron has 4 faces, all triangles, with 6 edges and 4 vertices — too few faces, and no rectangular faces at all.
- (d) 8 — A cuboid is a 3D shape with 6 rectangular faces, 12 edges and 8 vertices (corners). Counting the corners of a cuboid gives 8. A candidate who mixes up vertices with faces answers 6. A candidate who mixes up vertices with edges answers 12. A candidate who counts only the 4 corners of the top face, forgetting the 4 corners of the bottom face, answers 4. The correct number of vertices is 8.
- (a) 7 — Rearranging F + V − E = 2 gives F = 2 − V + E = 2 − 10 + 15 = 7. A candidate who works out E − V without the +2, giving 15 − 10, answers 5. A candidate who rearranges with a sign error, working out 2 + V − E = 2 + 10 − 15 = −3 and then drops the negative sign, answers 3. A candidate who adds all three numbers together, V + E + 2 = 10 + 15 + 2, answers 27, having used the wrong operation entirely. The correct number of faces is 7.
- (d) 0 — A vertex is a sharp corner point where edges meet. A cylinder has two curved, circular edges but no sharp corner points at all, so it has 0 vertices, making that the correct answer. '2' wrongly treats the two circular edges themselves as vertices, but an edge is not the same as a vertex. '4' overcounts by treating each circular edge as if it had two end-vertices, which does not apply to a continuous curved edge. '1' wrongly imagines the curved surface itself forming a single corner point, which it does not.
- (d) A prism — A prism has two identical, parallel polygon faces (its cross-section) joined by flat rectangular side faces, so the same cross-sectional shape runs all the way along its length — this description matches a prism. A pyramid instead narrows from one polygon base up to a single point (the apex), so it does not have two identical parallel faces. A cone has one curved surface and a single circular base narrowing to a point — no flat rectangular sides at all. A cylinder has two identical circular faces, but they are joined by a curved surface, not by flat rectangles.
- (b) No, it needs one more square — A closed cube has exactly 6 faces, so its net must be made of exactly 6 identical squares, arranged so each one unfolds to a separate face with none overlapping. This net has only 5 squares, so it is one square short and cannot be folded into a closed cube. Choosing 'Yes, it folds into a cube' ignores that a cube needs 6 faces, not 5. Choosing 'No, it has one square too many' miscounts in the wrong direction — 5 is one too FEW, not one too many. Choosing 'Yes, but only if two squares overlap' is not a valid net: a net's faces must not overlap when folded.
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