3D Shapes - Faces, Edges and Vertices: Edexcel GCSE Foundation maths revision notes

3-D shapes in the exam

A 3-D shape, or solid, is a shape that takes up space, such as a box or a ball. This page covers the names of solids and how to count their faces, edges and vertices.

In the exam you are usually asked to write down the name of a 3-D shape, or the number of faces or edges it has, for 1 mark each. You may also be asked to draw a sketch of a solid, such as a triangular prism, or to shade a face named by the letters at its corners.

A harder question, worth 2 marks, gives the length of the edges and asks for the total length of all the edges. Count the edges carefully first, then multiply.

Faces, edges and vertices

A face is a flat surface of a solid. An edge is a line where two faces meet. A vertex is a corner, where edges meet. The word for more than one vertex is vertices.

A cuboid has 6 faces, 12 edges and 8 vertices. A cube is a cuboid whose faces are all squares, so it has the same numbers.

Most diagrams of solids show the edges you cannot see, at the back, as dashed lines. Count those too. Tick each edge on the diagram as you count it, so you don't miss one or count one twice.

Names of solids

Cube: 6 square faces.
Cuboid: 6 rectangular faces, like a box.
Prism: two identical ends joined by rectangles. It is named after its ends, so a prism with triangles at the ends is a triangular prism.
Pyramid: a flat base with triangles that meet at a point at the top. It is named after its base, such as a square-based pyramid.

Cylinder: 2 flat circular faces, joined by one curved surface.
Cone: 1 flat circular face, and a curved surface that rises to a point.
Sphere: a ball shape, with one curved surface and no flat faces.

Always give the 3-D name. A solid with circles at both ends is a cylinder, not a circle, and a box shape is a cuboid, not a rectangle. Give one name only: an answer such as "cylinder cone" gets no mark.

Worked example 1: Naming two solids

Here are two 3-D shapes, A and B. Write down the mathematical name of each shape.

Shape A has a circle at each end, joined by one curved surface.
Your turn. How many flat faces does shape A have?

Your answer

Shape A has 2 flat faces, both circles, so it is a cylinder.
It is lying on its side, but it is still a cylinder. Write cylinder, not circle.
Shape B has a triangle at each end, joined by 3 rectangles. It does not come to a point, so it is a triangular prism, not a pyramid.
Answer: A is a cylinder and B is a triangular prism.

Worked example 2: Counting on a square-based pyramid

Here is a square-based pyramid. Work out how many faces, edges and vertices it has.

Faces: 1 square base and 4 triangles, so 5 faces.
Edges: there are 4 edges round the base, and 4 sloping edges that meet at the top.
Your turn. How many edges does the pyramid have altogether?

Your answer

4+4=8 edges
Vertices: 4 corners of the base and 1 at the top, so 5 vertices.
Answer: 5 faces, 8 edges and 5 vertices

Counting patterns for prisms and pyramids

A triangular prism has 2 triangles at its ends and 3 rectangles joining them, so 5 faces. It has 3 edges on each end and 3 joining the ends, so 9 edges. It has 3 vertices on each end, so 6 vertices.

To count the edges of a prism, count the edges round one end, then the same number round the other end, then the edges joining the two ends, one at each corner.
To count the edges of a pyramid, count the edges round the base, then the sloping edges, one from each corner of the base up to the top.

For a prism with a pentagon at each end: 5 edges round the front, 5 round the back and 5 joining them, so 5+5+5=15 edges. It has 10 vertices, so don't mix the two up.

Worked example 3: Naming a face by its corners

Here is a triangular prism. Some of its vertices are labelled. (a) Shade in the face PRTS. (b) Write down the number of faces of the prism.

A face is named by the letters at its corners, in order round the face. Find P, R, T and S on the diagram.
P and R are corners of the front triangle, and S and T are behind them. So PRTS is the sloping rectangle on the right.
Faces: there is a triangle at each end, PQR at the front and one at the back. This diagram does not show the edges you cannot see, but the faces at the back still count.
Your turn. There is a rectangle along each side of the triangle, joining the two ends. How many faces does the prism have altogether?

Your answer

2 triangles and 3 rectangles, so 2+3=5 faces.
Answer: (a) face PRTS is the sloping rectangle on the right, shaded (b) 5 faces

Worked example 4: Counting on a hexagonal prism

Here is a hexagonal prism. The dashed lines show the edges you cannot see. Work out the number of faces, edges and vertices of the prism.

Each end is a hexagon, with 6 sides.
Faces: 6 rectangles round the sides and 2 hexagons at the ends, so 8 faces.
Edges: 6 on the front hexagon, 6 on the back hexagon, and 6 joining them.
Your turn. How many edges does the prism have altogether?

Your answer

6+6+6=18 edges
Vertices: 6 on each hexagon, so 2\times6=12 vertices.
Answer: 8 faces, 18 edges and 12 vertices

Sketching a 3-D shape

You may be asked to draw a sketch of a solid. A sketch does not need to be accurate, but it must clearly be the right shape.

To sketch a prism, draw one end. Draw the same shape again, a little up and to the right. Then join each corner to the matching corner.

If you can, show the edges you cannot see as dashed lines. A prism has the same shape at both ends, so its lines never meet at a point: a shape that comes to a point is a pyramid.

Total length of the edges

When every edge of a solid is the same length, count the edges, then multiply by the length of one edge. A cube with edges of 5 cm has 12 edges, so the total length of its edges is 12\times5=60 cm.

When the edges are not all the same length, put them into groups of the same length. Work out the total for each group, then add the totals.

Write the number of edges in your working, with the word edges, before you multiply by a length. Then the examiner can see what you counted.

Worked example 5: Total length when every edge is the same

A gift box is a prism. Each end of the box is an octagon. Gold tape is stuck along every edge of the box. Every edge is 12.5 cm long. Work out the total length of gold tape.

Each end is an octagon, with 8 sides.
Edges: 8 round the front octagon, 8 round the back octagon, and 8 joining them, one at each corner.
Your turn. How many edges does the box have altogether?

Your answer

8+8+8=24 edges
Every edge is 12.5 cm long: 24\times12.5=300 cm
Answer: 300 cm

Worked example 6: Total length with two lengths of edge

A tent frame is a triangular prism, with a pole along every edge. Each end of the tent is an equilateral triangle with sides of 1.5 m. The tent is 2.4 m long. Work out the total length of the poles.

A triangular prism has 9 edges: 3 on each triangular end and 3 joining the ends.
The 2 ends have 6 edges altogether, each 1.5 m long: 6\times1.5=9 m
Your turn. The 3 edges joining the ends are each 2.4 m long. Work out their total length.

Your answer

3\times2.4=7.2 m
Total: 9+7.2=16.2 m
Answer: 16.2 m

Common mistakes

Giving a 2-D name for a solid. A solid with circles at both ends is a cylinder, not a circle, and a box shape is a cuboid, not a rectangle.
Writing two names at once. Give one name, such as cone, because an answer such as "triangle cone" gets no mark.
Missing the hidden edges. The dashed lines are edges too, so a triangular prism has 9 edges, not 6.
Mixing up edges and vertices. Edges are lines and vertices are corners, so for a total length of edges, multiply by the number of edges.
Giving every prism 6 faces. Only a cuboid has 6 faces, and a triangular prism has 5.
Drawing a pyramid instead of a prism. A prism has the same shape at both ends, joined by rectangles, while a pyramid comes to a point.