Common Solids
Geometry · Common Solids
Syllabus tag: KCSE | Mathematics | Form 1 | Topic 23 Common Solids
Lesson objectives
By the end of this topic, you should be able to:
- Identify and sketch common solids.
- Identify the vertices, edges and faces of common solids.
- State the geometric properties of common solids.
- Sketch and interpret the nets of solids.
Common Solids
A solid occupies space. It has length, width and height.
a) The three terms
A face is a flat or curved surface.
An edge is where two faces meet.
A vertex is a point where edges meet. The plural is vertices.
b) The common solids
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square pyramid | 5 | 8 | 5 |
| Tetrahedron | 4 | 6 | 4 |
| Cylinder | 3 | 2 | 0 |
| Cone | 2 | 1 | 1 |
| Sphere | 1 | 0 | 0 |
A prism has the same cross-section throughout its length.
A pyramid has a base and triangular faces meeting at a point called the apex.
c) Euler's formula
For any simple solid with flat faces:
V − E + F = 2
Here V is the vertices, E the edges and F the faces.
A cube gives 8 − 12 + 6 = 2. A square pyramid gives 5 − 8 + 5 = 2. A triangular prism gives 6 − 9 + 5 = 2.
This is a useful check. If your counts do not give 2, you have miscounted.
It applies only to solids with flat faces. A cylinder, cone and sphere have curved surfaces and do not obey it.
d) Nets
A net is the solid opened out flat. Every face appears once, at true size.
A cuboid's net shows six rectangles in three matching pairs.
The net must fold back into the solid, so faces must sit in positions that allow it. The same solid has several valid nets.
A cylinder unrolls into two circles and one rectangle.
The rectangle's length equals the circumference of the circle, 2πr. That single fact gives the curved surface area as 2πrh.
e) Surface area from a net
The surface area is the total of all the faces.
Working from a net makes this reliable, because every face is visible and flat.
For a cuboid l by w by h, the surface area is 2(lw + lh + wh).
For a closed cylinder, it is 2πr² + 2πrh.
f) Sketching solids
Draw the front face, then draw parallel edges going back, then join the corners.
Show hidden edges as dashed lines. That is the convention, and marks are given for it.
g) Where this is used
Packaging design, where nets decide how much card is cut. Sheet metal work. Estimating paint for a tank. Any object that must be built from flat material.
Words to know
- Face -- a flat or curved surface of a solid.
- Edge -- the line where two faces meet.
- Vertex -- a corner where edges meet.
- Net -- the flat pattern that folds to form a solid.
- Apex -- the point where the triangular faces of a pyramid meet.
:::checkpoint Check yourself
- How many edges has a triangular prism?
- Use Euler's formula to check a tetrahedron.
- Why does a cylinder not obey Euler's formula?
- What shapes make up the net of a cylinder? :::
Bridge to practice
The exercises begin with faces, edges and vertices, move through prisms, pyramids and Euler's relation, and finish with nets and surface area. Whenever you count the parts of a solid, check the three numbers against F + V − E = 2 before writing them down.