Polyhedra and their parts

A polyhedron (plural polyhedra) is a solid whose surface is made entirely of flat polygonal faces. Cubes, pyramids, dice, and rooms are all polyhedra. Footballs and globes are not , they have curved surfaces.

Concept

Three vocabulary words.

• Face: a flat polygon that bounds the solid.
• Edge: a line segment where two faces meet.
• Vertex: a point where three or more edges meet.

A cube has $6$ faces (each a square), $12$ edges, $8$ vertices.

Types of polyhedra.

• Prism: two parallel congruent polygon faces (the "bases") connected by rectangles. Examples: a triangular prism (toblerone shape), a rectangular prism (box, cuboid).
• Pyramid: a polygon base and triangular faces meeting at a single apex. Examples: square pyramid (Egyptian-style), triangular pyramid (tetrahedron).
• Regular polyhedron: every face is the same regular polygon, and every vertex looks the same. Only five of these exist , the famous Platonic solids.

The five Platonic solids.

These were known to the Greeks and are named after Plato, who described them in the Timaeus around $360$ BCE. He linked them to the four classical elements and the cosmos.

Convex vs concave. A polyhedron is convex if any line segment with endpoints inside it lies entirely inside. A "star polyhedron" (a star shape pulled into 3D) is concave.

Cross sections. Cut a polyhedron with a flat plane. The intersection is a polygon. The cross sections of a cube can be triangles, quadrilaterals, pentagons, or even regular hexagons , depending on the angle of the cut.

Worked examples

Example 1. How many faces, edges, vertices does a triangular prism have?

• Faces: $5$ (two triangles + three rectangles).
• Vertices: $6$ (three per triangular base).
• Edges: $9$ (three per base + three connecting).

Example 2. A pyramid has a hexagonal base. How many faces and vertices?

• Faces: $7$ (one hexagonal base + six triangular sides).
• Vertices: $7$ (six on the base + one apex).
• Edges: $12$ (six on base + six along sides).

Example 3. Verify Euler's formula for a cube.

• $F = 6$, $V = 8$, $E = 12$.
• $F + V - E = 6 + 8 - 12 = 2$. ✓

Example 4. A regular tetrahedron has $4$ faces, $4$ vertices. How many edges?

• $F + V - E = 2 \Rightarrow E = F + V - 2 = 6$. ✓

Try it yourself

1. Count $F, V, E$ for a triangular pyramid (tetrahedron).
2. Count for a square pyramid (Egyptian style).
3. A prism has a pentagonal base. Find $F, V, E$.
4. List the five Platonic solids and their face shapes.
5. A cuboid: how many faces meet at each vertex?
6. Sketch a cross-section of a cube that is a regular hexagon.
7. Verify Euler's formula for an octahedron.
8. Look up a "rhombic dodecahedron". What are $F, V, E$?

Activity

Polyhedron hunt. Find five polyhedra in your kitchen or classroom (cubes of sugar, packaging, ice trays, pencils, books). For each, count $F, V, E$ and verify Euler's formula. You will find it works every single time , even on the strangest shape you can find.