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.
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Answer key: Properties of 3D shapes worksheet — GCSE Foundation
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- (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.
- (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) 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.
- (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.
- (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) 12 — A cube has 6 square faces, 12 edges and 8 vertices. The edges are the straight lines where two faces meet: 4 edges around the top face, 4 edges around the bottom face, and 4 vertical edges joining them, giving 4 + 4 + 4 = 12. A candidate who counts the vertices instead of the edges answers 8. A candidate who counts the faces instead answers 6. A candidate who counts only the edges around the top face, forgetting the bottom face and the vertical edges, answers 4. The correct number of edges is 12.
- (b) 3 — A cuboid with all different edge lengths has three planes of symmetry: one parallel to each pair of opposite faces, cutting the solid exactly in half. Choosing 9 is the number of planes of symmetry a CUBE has (where all edges are equal) — this cuboid's edges are all different, so it has fewer. Choosing 1 counts only one of the three planes and forgets the other two, each parallel to a different pair of faces. Choosing 6 double-counts each of the three planes, as if counting each one from both sides.
- (b) Cuboid — not necessarily a cube — A solid with 6 faces, 12 edges and 8 vertices in which every face is a rectangle is a cuboid, but nothing here confirms that all the edges are the same length, so the box could be a cube or a non-cube cuboid; the most that can be concluded is that it is a cuboid, making 'Cuboid — not necessarily a cube' correct. 'Cube — only a cube fits this' is wrong because a cube is just one particular cuboid; a general cuboid with different length, width and height has exactly the same face, edge and vertex counts and rectangular faces. 'Triangular prism' is wrong because a triangular prism has 5 faces, 9 edges and 6 vertices, and two of its faces are triangles, so it matches neither the counts nor the face shape. 'Not enough information' is wrong because rectangular faces with these counts do pin the solid down to the cuboid family, even though they cannot pin down a cube specifically.
- (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.
- (c) 18 — Rearranging F + V − E = 2 gives E = F + V − 2. Substitute F = 8 and V = 12: 8 + 12 − 2 = 18 edges. Choosing 20 comes from adding the faces and vertices but forgetting to subtract the 2 (8 + 12 = 20). Choosing 22 comes from adding the 2 instead of subtracting it (8 + 12 + 2 = 22). Choosing 16 comes from subtracting 2 twice by mistake (8 + 12 − 2 − 2 = 16).
- (a) 2 — A cone has one flat face — the circular base — and one curved surface, which is counted as a single face. That gives a total of 2 faces. A candidate who forgets the circular base and counts only the curved surface answers 1. A candidate who mistakenly splits the curved surface into two faces answers 3. A candidate who thinks a cone has no flat faces at all answers 0. The correct number of faces is 2.
- (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.
- (b) Hexagon — Cutting straight across a prism, at right angles to its length, always gives a cross-section that is the same shape as its end faces. The end faces of a hexagonal prism are hexagons (6-sided), so the cross-section is a hexagon. Choosing Pentagon comes from miscounting the sides of the hexagonal end as five instead of six. Choosing Rectangle comes from cutting along the LENGTH of the prism instead of across it, which gives a rectangular face, not the cross-section asked for. Choosing Triangle comes from confusing a hexagonal prism with a triangular prism.
- (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) 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.
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