Mathematics (Core)Exam code: 0580
Symmetry and Shapes
Geometry
Naming solids
- A prism has the same cross-section all the way along, and it is named after that cross-section
- A cube has a square cross-section and a cuboid a rectangular one
- A triangular prism and a hexagonal prism are named the same way
- A cylinder behaves like a prism whose cross-section is a circle
- A pyramid stands on a base, with triangular sides rising to meet at a single point above it, and takes its name from that base
- A pyramid on a triangular base is a tetrahedron
- A cone is like a pyramid on a circular base, and a sphere is a ball
Faces, edges and vertices
- A face is a flat surface, an edge joins two faces, and a vertex is a corner; the plural of vertex is vertices
- A curved surface, such as the side of a cylinder, counts as a surface rather than a face
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 squares | 12 | 8 |
| Cuboid | 6 rectangles, in 3 matching pairs | 12 | 8 |
| Triangular prism | 2 triangles and 3 rectangles | 9 | 6 |
| Square-based pyramid | 1 square and 4 triangles | 8 | 5 |
| Tetrahedron | 4 triangles | 6 | 4 |
| Cylinder | 2 circles, plus a curved surface | 2 | 0 |
| Cone | 1 circle, plus a curved surface | 1 | 1 |
| Sphere | curved surface only | 0 | 0 |
- For a solid with flat faces and no hole through it, the three counts satisfy faces − edges + vertices = 2, which is a quick way to check a count
Worked example
Counting faces, edges and vertices
Write down the number of faces, edges and vertices of a pyramid with a pentagonal base and of a hexagonal prism.
Solution:
- For the pyramid, count the flat surfaces first
- One pentagonal base and a triangle on each of its five sides gives 6 faces
- The base has 5 edges, and 5 more rise from its corners to the apex, giving 10 edges
- The 5 corners of the base plus the apex give 6 vertices
- For the hexagonal prism, the two hexagonal ends plus one rectangle for each of the six sides give 8 faces
- Six edges round each hexagon plus six joining them gives 18 edges
- Six corners on each hexagon gives 12 vertices
- Check each with Euler's relationship, faces + vertices − edges = 2: 6 + 6 − 10 = 2, and 8 + 12 − 18 = 2
Nets
- A net is the flat shape that folds up to make a solid, so it shows every face laid out
- The area of a net equals the surface area of the solid, which is why nets are worth drawing for surface-area questions
- A solid has more than one possible net, and a cube has eleven, so an arrangement different from yours may still be correct
- The net of a cylinder is two circles and a rectangle whose length equals the circumference of those circles
- Check a proposed net by asking whether the faces would overlap or leave a gap when folded
Exam tip
A part-finished net still scores
Drawing or completing a net was set in 10 of the 22 papers, always for 3 marks. The marks are given face by face: an unfinished net still earns one or two marks for the correct faces it adds in their correct places. So if the whole net will not come, still draw every face you are sure of, joined along an edge of the same length.
Worked example
Completing a net of a cuboid
A cuboid measures 6 cm by 3 cm by 2 cm. One of its 6 cm by 3 cm faces has been drawn on a 1 cm² grid. Complete a net of the cuboid.
Solution:
- A cuboid has 6 faces in 3 matching pairs: here two 6 by 3 rectangles, two 6 by 2 and two 3 by 2
- Use the drawn face as the base, with its 6 cm edges at the top and bottom
- Join a 6 by 2 rectangle to each 6 cm edge of the base: these fold up to make the two long sides
- Join the second 6 by 3 rectangle to the outer 6 cm edge of the upper side: it folds over to make the lid
- Join a rectangle 2 squares wide and 3 tall to each 3 cm edge of the base: these fold up to make the two ends
- Check: there are 6 faces, two of each size, and every pair of edges that meet when it is folded are the same length
- The net is a column of four rectangles 6 squares wide, 3, 2, 3 and 2 squares tall from the top, with a 2 by 3 rectangle on each side of the base