GCSE-readyProperties of 3-D shapes and cylinders
Prisms, pyramids, cylinders, cones and spheres are everywhere, from tins to tents. Count their parts and work out cylinder volumes and surface areas.
Faces, edges and vertices
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube or cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square-based pyramid | 5 | 8 | 5 |
| Tetrahedron (triangular pyramid) | 4 | 6 | 4 |
Patterns
A pyramid with an n-sided base has n + 1 faces, 2n edges and n + 1 vertices.
For solids with flat faces and no holes, Euler's formula holds: F + V − E = 2.
Cylinders, cones and spheres have curved surfaces. A sphere has one curved surface; a cone has one flat face and one curved surface.
Cylinders
Volume = πr2h
Curved surface area = 2πrh (unroll it into a rectangle: its width is the circumference 2πr)
Total surface area = 2πr2 + 2πrh (two circles plus the curved part)
A cylinder has radius 3 cm and height 8 cm. Find its total surface area to 1 decimal place.
- Two circles: 2 × π × 32 = 18π
- Curved surface: 2 × π × 3 × 8 = 48π
- Total: 66π = 207.345...
Answer: 207.3 cm2
Prisms: n + 2 faces, 3n edges, 2n vertices. Pyramids: n + 1 faces, 2n edges, n + 1 vertices. F + V − E = 2. Cylinder volume πr2h, curved surface 2πrh.
- Volume
- The amount of space inside a 3-D shape. Measured in cubic units such as cm^{3}.
- Prism
- A 3-D shape with the same cross-section all the way through, such as a cuboid or a triangular prism.
- Polygon
- A flat shape with straight sides, such as a pentagon (5 sides) or hexagon (6 sides).
- Formula
- A rule linking two or more variables, such as A = lw for the area of a rectangle.
- Surface area
- The total area of all the faces of a 3-D shape.
- Area
- The amount of flat space a 2-D shape covers. Measured in square units such as cm^{2} or m^{2}.
- Not “Cylinder”. A cylinder has two flat circular faces.
- Not “Sphere”. A sphere has no flat faces.
- Not “Triangular prism”. A prism has only flat faces.
Try two quick questions
See if it has sunk in. Answers are marked straight away.
2 more questions
Try each one, then tap to check.
How many edges does a pentagonal prism have?
Show the answer
Answer: 15
Two pentagon ends with 5 edges each, plus 5 joining edges: 15.
How many vertices does a square-based pyramid have?
Show the answer
Answer: 5
4 vertices around the base plus 1 at the top: 5.
Check you have got it
Answer 7 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.
Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.