3D Shapes - Faces, Edges and Vertices

Look at the shape of the two ends and the side that joins them.

Remember to count the faces you cannot see, as well as the ones you can.

Count the corners and edges on both triangular ends first, then the edges that join the two ends. Include the dashed ones.

A vertex is a corner. Count the corners of the base, then the point at the top.

An edge is a line where two faces meet. Count the edges round the top, the edges round the bottom and the edges going up the sides. Include the dashed ones.

Every face of this pyramid is a triangle, including the base. Count the edges around the base first, then the edges going up to the top.

Part (a)

1. The base is a triangle: 1 face.
2. There is a sloping triangular face above each side of the base: 3 faces.
3. 1+3=4 faces.

Part (b)

1. The base has 3 edges, including the dashed one.
2. There are 3 edges from the corners of the base up to the top.
3. 3+3=6 edges.

A prism has the same shape at both ends. A pyramid comes to a point.

1. The two triangles are the two ends, and the three rectangles join them together.
2. A solid with the same triangle at each end is a triangular prism.
3. (A square-based pyramid also has 5 faces, but they are 1 square and 4 triangles.)

A flat face is a face you could stand the solid on, like a circle. A curved surface is not flat.

Part (a)

1. The cylinder has a circle at the top and a circle at the bottom.
2. These are its 2 flat faces. The side is one curved surface.

Part (b)

1. The cone has one circle, at its base.
2. So it has 1 flat face. The rest of the cone is one curved surface that comes to a point.

Give the name of the solid, not the name of a flat shape. Look at the shape of the faces of A, and whether B has any flat faces at all.

Part (a)

1. Shape A is a box shape with 6 rectangular faces, so it is a cuboid.
2. It is not a cube, because its faces are not all squares. A rectangle is a flat shape, not a solid.

Part (b)

1. Shape B is a ball shape, with one curved surface and no flat faces, so it is a sphere.
2. A cylinder and a cone both have flat circular faces. A circle is a flat shape, not a solid.

The two ends are pentagons. Count the corners and edges on both ends, then the edges that join the two ends.

Part (a)

1. There are 2 pentagon faces, one at each end.
2. There is a rectangle for each side of the pentagon: 5 rectangles.
3. 2+5=7 faces.

Part (b)

1. Each pentagon has 5 edges: 5+5=10 edges.
2. There are 5 edges joining the two ends.
3. 10+5=15 edges.

Part (c)

1. Each pentagon has 5 vertices (corners).
2. 5+5=10 vertices.

A face is named by the letters at its corners, going round the face. Each letter must be joined to the next one by an edge. For the number of faces, count the triangles and the rectangles.

Part (a)

1. A to C, C to F, F to D and D back to A are all edges of the prism, so ACFD is a face. It is the sloping rectangle on the right.
2. In ABFD and ACED, two of the corners are not joined by an edge (B to F, and C to E), so they are not faces.
3. B, C and D do not all lie on one face, so BCD is not a face.

Part (b)

1. There are 2 triangles, ABC at the front and DEF at the back.
2. There are 3 rectangles joining them: ACFD, ABED and BCFE.
3. 2+3=5 faces.

Think about the base first, then the edges and faces that go up to the point at the top.

Part (a)

1. The base has 8 edges.
2. There is one edge from each of the 8 corners of the base up to the top.
3. 8+8=16 edges.

Part (b)

1. There is a triangular face above each side of the base: 8 triangles.
2. The base is 1 more face.
3. 8+1=9 faces.

A cuboid has 12 edges. How many of them are 12 cm long, how many are 5 cm and how many are 3 cm?

1. A cuboid has 12 edges: 4 of each length.
2. 4\times12=48 cm, 4\times5=20 cm and 4\times3=12 cm.
3. 48+20+12=80 cm.

A prism has the same shape at both ends, with the matching corners joined. A shape whose edges meet at a point at the top is a pyramid.

1. Sketches A and B come to a point at the top, so they are pyramids: A has a square base and B has a triangle base.
2. Sketch D has rectangles at both ends, so it is a cuboid.
3. Sketch C has the same triangle at the front and at the back, with the corners joined, so it is a triangular prism.

How many corners does a cube have? How many edges does it have? Which of these has Amir used?

Part (a)

1. A cube has 8 vertices (corners), 12 edges and 6 faces.
2. Amir multiplied by 8, which is the number of vertices, not the number of edges.

Part (b)

1. A cube has 12 edges.
2. 12\times9=108 cm.

Work out the faces, edges and vertices of each solid. Only one solid matches all three numbers.

1. Pentagonal prism: 7 faces, 15 edges, 10 vertices. The faces match, but not the edges or vertices.
2. Hexagonal prism: 8 faces, 18 edges, 12 vertices.
3. Pyramid with a hexagon base: 6 triangles +1 base =7 faces, 6+6=12 edges, 6+1=7 vertices.
4. Pyramid with a pentagon base: 6 faces, 10 edges, 6 vertices.
5. The solid is a pyramid with a hexagon base.

Count the edges round the front hexagon, round the back hexagon, and the edges joining the two ends. Then multiply by the length of one edge.

1. Each hexagon has 6 edges, so the two ends have 6+6=12 edges.
2. There are 6 more edges joining the two ends, one at each corner.
3. The prism has 12+6=18 edges.
4. Total length =18\times4.5=81 cm.

Count the edges round the square base and the sloping edges up to the top separately. Which length goes with each group?

1. A square-based pyramid has 8 edges: 4 round the square base and 4 sloping edges up to the top.
2. The 4 edges of the base are each 7 cm long: 4\times7.
3. The 4 sloping edges are each 12 cm long: 4\times12.
4. So the total length is 4\times7+4\times12=28+48=76 cm.
5. 5 is the number of vertices, not edges, and 8\times7+8\times12 counts every stick at both lengths.

Count the edges of the pentagons and the edges that join the two ends separately. Remember that 1 m =100 cm.

1. The two pentagons have 5+5=10 edges, each 4 cm: 10\times4=40 cm.
2. There are 5 edges joining the two ends, each 15 cm: 5\times15=75 cm.
3. Total wire used =40+75=115 cm.
4. 2 m =200 cm, so she has 200-115=85 cm left.

A triangular prism has 9 edges. Put them into groups of the same length: how many are 35 cm, how many are 40 cm and how many are 90 cm?

1. A triangular prism has 3 edges on each end and 3 edges joining the ends: 9 edges.
2. Each end has two 35 cm sides, so there are 4 edges of 35 cm: 4\times35=140 cm.
3. Each end has one 40 cm side, so there are 2 edges of 40 cm: 2\times40=80 cm.
4. The 3 edges joining the ends are each 90 cm: 3\times90=270 cm.
5. Total length =140+80+270=490 cm.