These questions ask you to calculate or work backwards from a volume, compare how much two solids hold, or count faces, edges and vertices. Begin by deciding whether the question concerns space inside the solid or features on its boundary.
Use three dimensions for volume
For a cuboid, multiply all three perpendicular dimensions. A box measuring 6 by 4 by 2 has volume 48 cubic units. For a cube, the three dimensions are equal, so an edge of 5 gives 5 times 5 times 5, or 125 cubic units. To recover a missing height, divide the volume by the base area: a volume of 96 over a base of 8 by 3 gives a height of 4. Compare solids by comparing these products, not by looking at one edge.
Count a solid systematically
| Solid pattern | Counting method |
|---|---|
| Cube or cuboid | There are 6 faces, 12 edges and 8 vertices. |
| Prism | Count the two matching end faces, then one joining face for every edge of an end. Count both sets of end corners, and include one joining edge for every corner on an end. |
| Pyramid | Count the base and every triangular face rising from it. Count every base corner plus the apex as vertices, and include both the base edges and the edges running to the apex. |
Avoid these solid-geometry traps
| Tempting route | How to reject it |
|---|---|
| Give the corner count when edges are requested | Name the feature before counting. Edges are line segments; vertices are points. |
| Multiply only two dimensions for volume | Two dimensions measure one face. The third dimension supplies the layers behind that face. |
| Find a missing height by dividing by only one base side | Multiply both base dimensions first, then divide the volume by that base area. |
| Count only the sloping faces of a pyramid | The base closes the solid and is a face too. Include it before totaling. |
| Use total surface area as volume | A covering is measured in area units such as cm²; the space inside is measured in cubic units. |
| Count only the two end outlines of a prism | Add the joining edges that run from one matching end to the other. |
Try it
A cuboid measures 5 centimetres long, 3 centimetres wide and 2 centimetres high. Enter the number of cubic centimetres in its volume.
A square-based pyramid has a square base and four triangular faces that meet at a single point above it. Select all the statements that are true about this solid.
- It has 5 vertices: the four corners of the base together with the point where the triangles meet.
- It has only 4 vertices, one for each corner of its square base.
- It has 5 faces in total, one square and four triangles.
- It has 8 vertices, one for each of its edges.
A cuboid measures 8 centimetres by 3 centimetres by 3 centimetres, and a cube has edges of 4 centimetres. Which solid holds more, and for what reason?
- The cube, because its edges are all equal and equal edges pack together most tightly.
- The cuboid, because 8 times 3 times 3 is 72 while 4 times 4 times 4 is 64.
- They hold exactly the same, because both are built from 12 edges and 6 faces.
- The cube, because its edge of 4 centimetres beats the shortest edge of the cuboid, which is only 3 centimetres.