3D Shapes - Faces, Edges, Vertices and Nets

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.

Imagine folding each diagram up. A net of a cube must make six faces with no two squares landing on the same face.

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.

Look at the shapes in the net. Which face would the solid stand on, and where do the other faces meet when they fold up?

1. The net has 1 square and 4 triangles.
2. The square is the base. The four triangles fold up from its sides and meet at a point at the top.
3. A solid with a square base and four triangular faces meeting at a point is a square-based pyramid.

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.

Two faces that are next to each other on the net can't be opposite. Try folding the net in your head, one square at a time.

1. Imagine C as the bottom of the cube.
2. B folds up to make the back. A is joined to the left edge of B, so it folds round to make the left side.
3. D folds up to make the right side, E folds over from D to make the top, and F folds up from D to make the front.
4. A is the left side and D is the right side, so face D is opposite face A.

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.

Imagine folding the long row of squares round. What happens to the squares at the two ends?

1. Fold the row round: the second square makes one side, the third the bottom, the fourth the other side.
2. The first and fifth squares both fold up to make the top, so they would overlap.
3. That leaves only 5 different faces, so one face of the cube would be missing.
4. The diagram does have 6 squares, all the same size, so the other reasons are wrong.

Each side of the hexagon has a rectangle joined to it.

Part (a)

1. A hexagon has 6 sides.
2. There is one rectangle along each side of the hexagon, so there are 6 rectangles.

Part (b)

1. There is a hexagon at each end of the prism.
2. So there are 2 hexagons.

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.

Choose one square as the bottom of the cube and fold the other squares up round it. Track where the square with P on it ends up.

1. Use the square in the second column, second row as the bottom of the cube.
2. The square to its left folds up to make the left side. The square to its right folds up to make the right side, and the last square in the row folds over to make the top.
3. The top folds over from the right, so its far edge, which includes P, meets the top edge of the left side. P meets the bottom left corner of the row.
4. The square below the bottom folds up to make the front. Its bottom left corner, B, meets that same corner.
5. So point B meets point P.

Fold each diagram in your head. A diagram is not a net if two squares would end up on the same face of the cube.

1. A: fold the row of four round to make four sides. The square at the top left and the square at the bottom right fold over to make the two ends. It is a net.
2. B: two rows of three squares. Folding it, squares overlap and two faces are missing. It is not a net.
3. C: the first three squares make the bottom, one side and the top; the last three make the other three faces. It is a net.
4. D: the two squares on top are above the two ends of the row of four. They both fold onto the same face, so it is not a net.
5. The nets are A and C.

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.

Find the length of every edge around the outside of the net. Each edge is 8 cm, 5 cm or 3 cm.

1. Work round the outside of the net, edge by edge.
2. Top face: 3+8+3=14 cm. Bottom face: 3+8+3=14 cm.
3. Row of four faces: along the top and bottom, the parts not covered by the top and bottom faces are 3+3+8=14 cm each, so 28 cm. The two ends are 5 cm each, so 10 cm.
4. Perimeter =14+14+28+10=66 cm.