Exploring Some Geometric Themes · Lesson 5 of 13
Making Solids and Counting Their Parts
“Recognise prisms and pyramids and derive their face, edge, and vertex counts from their structure.”
Identify faces, straight edges, and vertices of polyhedra. Distinguish prisms, pyramids, and curved solids. Derive counting rules for prisms and pyramids with polygon bases. Use diagrams and models to include hidden parts in a count.
A box looks simple until you try to count its parts from a single drawing. Some faces point away from you, and some edges are hidden behind the solid. A reliable count comes from understanding how the object is put together.
We will group solids into families. Once you understand what repeats within a family, you can count the parts of a solid with a ten-sided base without needing a detailed drawing of every edge.
Faces, Edges, and Vertices
A cube has six flat square faces. Two neighbouring faces meet along an edge, and three edges meet at each vertex. A vertex is a corner point; the plural is vertices. A polyhedron is a closed solid surface made from flat polygon faces.
A cube has twelve edges and eight vertices. A typical drawing shows only three faces clearly, but the solid still has six. Dashed lines can represent hidden edges, yet a drawing that omits them does not remove them from the object.
A solid bounded by flat polygon faces that meet along straight edges.
A cuboid also has six faces, twelve edges, and eight vertices, although its rectangular faces need not be squares. A parallelepiped has six parallelogram faces and may look like a box that has been pushed sideways. Its face shapes change, but these counts remain the same.
Problem
Explain the twelve edges and eight vertices of a cube by grouping them.
- 1.The top square has four edges, and the bottom square has another four.
- 2.Four upright edges join corresponding top and bottom vertices, giving 4 + 4 + 4 = 12 edges.
- 3.The top contributes four vertices and the bottom contributes four different vertices, giving 8 in total.
Prisms
A prism has two congruent polygon bases in parallel planes, joined at corresponding vertices. Its other faces are parallelograms. In a right prism the joining edges are perpendicular to the bases, so the side faces are rectangles.
The base does not have to be the face touching the table. The two matching end faces determine which kind of prism it is. A triangular prism has triangular bases; a pentagonal prism has pentagonal bases. Turning either solid onto another face does not change its family.
A polyhedron with two congruent parallel polygon bases and parallelogram side faces joining corresponding base edges.
Let each base have n sides. The two bases contribute 2 faces, and every base edge has one side face attached, giving n more. Each base has n vertices, giving 2n vertices altogether. For edges, count n around one base, n around the other, and n joining corresponding vertices.
Problem
Find the number of faces, edges, and vertices of a decagonal prism.
- 1.A decagon has n = 10 sides. There are two decagonal bases and ten side faces.
- 2.Therefore F = 10 + 2 = 12 faces.
- 3.Edges come in three groups of ten, so E = 30. Vertices come in two groups of ten, so V = 20.
- 4.The result comes from the structure, even if the back edges would be difficult to see in a sketch.
Pyramids
A pyramid has one polygon base and triangular side faces meeting at one point called the apex. Join the apex to each base vertex. These joining edges separate neighbouring triangular faces.
For an n-sided base there are n triangular side faces and one base, so F = n + 1. The base gives n vertices and the apex adds one, so V = n + 1. The n base edges and n edges to the apex give E = 2n. The apex may be directly over the centre in a symmetric model, but the counting does not depend on that placement.
A polyhedron with one polygon base and triangular side faces that meet at a common apex.
A triangular pyramid has four triangular faces and is called a tetrahedron. A regular tetrahedron has four congruent equilateral faces. The word tetrahedron alone tells us the face count; regular gives the additional equality conditions.
Problem
Compare a pentagonal prism with a pentagonal pyramid.
- 1.For each solid, n = 5 because the base polygon is a pentagon.
- 2.The prism has F = 7, E = 15, and V = 10. It has two pentagonal bases.
- 3.The pyramid has F = 6, E = 10, and V = 6. It has one pentagonal base and one apex.
- 4.Sharing the same base shape does not give the same structure or counts.
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Triangular prism | 5 | 9 | 6 |
| Square prism | 6 | 12 | 8 |
| Pentagonal prism | 7 | 15 | 10 |
| Triangular pyramid | 4 | 6 | 4 |
| Square pyramid | 5 | 8 | 5 |
| Decagonal pyramid | 11 | 20 | 11 |
Each physical edge belongs to two neighbouring faces. Adding all face-side counts directly counts every edge twice. Also, the apex is one shared vertex, not a separate new vertex for every triangular face.
Curved Solids and Flat-Faced Solids
A cylinder has two circular flat bases and one curved surface. A cone has one circular base, one curved surface, and an apex. A sphere has a continuous curved surface and no vertices. These solids are not polyhedra because their boundaries are not made entirely of polygon faces.
In elementary descriptions, the circular boundary of a cone or cylinder is sometimes called a curved edge. Here the prism and pyramid formulas count straight polyhedral edges only. A circle is not a polygon with an ordinary finite number of sides, so substituting a made-up n for a cylinder or cone does not make these formulas applicable.
Build the Structure Before Counting
Use sticks and small connectors, or rolled paper strips and tape, to make a frame. This makes hidden edges easier to inspect than in a filled drawing.
- Make two matching pentagon frames and connect corresponding corners to build a prism frame.
- Count the two groups of base edges and the five joining edges.
- Make a single pentagon and connect all its corners to one point to build a pyramid frame.
- Turn both frames, recount, and explain why the totals do not change with the viewpoint.
The prism frame has 15 edges and 10 vertices; the pyramid frame has 10 edges and 6 vertices. Imagine adding paper faces to close each frame. Face counts then follow from the spaces that must be covered.
Quiz
How many side faces does an eight-sided prism have?
A pyramid has a six-sided base. How many vertices does it have?
Which description identifies a triangular prism?
Why is a cylinder not a polyhedron?
A prism has 18 edges. How many sides does each base have?
Practice Problems
- Find F, E, and V for a heptagonal prism.
- Find F, E, and V for a decagonal pyramid.
- A pyramid has 16 edges. Identify its base polygon and find its face count.
- Someone counts 24 edges on a cube by adding four sides for each of six faces. Explain the error.
- A solid has six faces. Must it be a cube? Give two reasons for your answer.
1. A heptagon has n = 7. 2. F = 7 + 2 = 9; E = 3 × 7 = 21; V = 2 × 7 = 14.
Key Takeaways
Faces, edges, and vertices belong to the complete solid, including hidden parts. An n-sided prism has n + 2 faces, 3n edges, and 2n vertices. An n-sided pyramid has n + 1 faces, 2n edges, and n + 1 vertices. A tetrahedron is a triangular pyramid; regularity adds equal-shape conditions. Curved solids do not follow the finite polygon counting rules for prisms and pyramids.