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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Properties of 3D shapes worksheet — GCSE Foundation
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- 1.How many vertices does a cuboid have?
- 2.A solid has 10 vertices and 15 edges. Using the formula F + V − E = 2, work out how many faces it has.
- 3.In this question, a curved surface counts as a face. How many faces does a cone have?
- 4.How many edges does a cube have?
- 5.A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles and the other three are rectangles. Which solid is it?
- 6.A hexagonal prism has 8 faces and 12 vertices. Using F + V − E = 2, work out how many edges it has.
- 7.A cuboid has three different edge lengths: a length, a width and a height, all different from each other. How many rectangles make up its net in total?
- 8.A tent is shaped like a triangular prism: two triangular end faces and three rectangular faces. One of the rectangular faces lies flat on the ground as the groundsheet. How many of the tent's faces are above the ground, not touching the ground?
- 9.A square-based pyramid has a square base and four identical triangular sloping faces, with its apex directly above the centre of the base. How many planes of symmetry does it have?
- 10.How many faces does a cuboid have?
- 11.A triangular prism has two triangular end faces and three rectangular faces joining them. How many faces does it have in total?
- 12.A cylindrical drum and a cuboid box are being wrapped for a display. A shop assistant wraps only the flat faces of each item, not any curved surface. How many flat faces in total does the assistant wrap across both items?
- 13.How many edges does a triangular prism have?
- 14.A solid has 5 faces: one square face and four triangular faces, which all meet at a single point above the square. What is the name of this solid?
- 15.A solid has 6 faces and 8 vertices. Using the formula F + V − E = 2, work out how many edges it has.
Answer key
- (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.
- (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) 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.
- (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.
- (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) 6 — A cuboid has six faces in total: a top, a bottom, a front, a back, and two ends — each one is a rectangle in the net, giving six rectangles altogether. Choosing 3 counts only the three PAIRS of congruent rectangles (top/bottom, front/back, two ends) rather than all six individual faces. Choosing 5 forgets one face, as if the net were missing its lid. Choosing 12 is the number of edges of a cuboid, not the number of rectangles in its net.
- (b) 4 — A triangular prism has 5 faces in total: 2 triangular ends and 3 rectangular faces. One rectangular face is the groundsheet, lying on the ground, so the other 5 − 1 = 4 faces are above the ground. A candidate who forgets one of the triangular ends when counting the remaining faces answers 3. A candidate who forgets to subtract the groundsheet at all answers 5. A candidate who only counts the two sloped rectangular faces, forgetting the two triangular ends, answers 2. The number of faces above the ground is 4.
- (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.
- (c) 6 — A cuboid has six flat faces: a top, a bottom and four sides. Count each flat surface once: top, bottom, front, back, left, right — six faces in total, so the answer is 6. Choosing 8 counts the vertices (corners) instead of the faces. Choosing 12 counts the edges instead of the faces. Choosing 4 counts only the four side faces and forgets the top and the bottom.
- (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.
- (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 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.
- (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) 12 — Method: rearrange F + V − E = 2 so that E is on its own: E = F + V − 2. Working: E = 6 + 8 − 2 = 12. A student who answers 14 has added F and V but forgotten to subtract 2 at all. A student who answers 16 has added 2 instead of subtracting it. A student who answers 10 has subtracted 2 twice by mistake. Answer: 12 edges.
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