What is a polyhedron? Faces, edges and vertices
A polyhedron (plural: polyhedra) is a closed solid whose whole surface is made of flat polygons. Each flat polygon is a face. Two faces meet along a straight line called an edge. Edges meet at a corner point called a vertex (plural: vertices).
A ball or a can is not a polyhedron, because part of its surface is curved.
A polyhedron is convex if the segment joining any two of its points stays inside it (no dents). A diagonal of a polyhedron joins two vertices that are not on the same face. The faces that meet at one vertex form a polyhedral angle; the angle between two faces along an edge is a dihedral angle.
Key fact: at every vertex of a convex polyhedron, the face angles add up to less than 360°. If they made exactly 360°, the faces would lie flat.
Prisms and parallelepipeds
A prism has two equal, parallel polygons as bases. The side faces (lateral faces) are parallelograms. The prism is named by its base: triangular prism, pentagonal prism and so on.
- Right prism: side edges are perpendicular to the bases, so side faces are rectangles.
- Oblique prism: side edges lean.
- Regular prism: a right prism whose base is a regular polygon.
With an n-sided base: F = n + 2, E = 3n, V = 2n.
A parallelepiped is a prism whose bases are parallelograms: all 6 faces are parallelograms, and opposite faces are equal and parallel. A cuboid (rectangular box) is a right parallelepiped with rectangle faces; its space diagonal is d = √(a² + b² + c²). A cube is a cuboid with all edges equal: d = a√3.
Pyramids, regular pyramids and frustums
A pyramid has one polygon base and triangular side faces that meet at one point, the apex. The height is the perpendicular distance from the apex to the base.
With an n-sided base: F = n + 1, E = 2n, V = n + 1.
A regular pyramid has a regular polygon base and its apex directly above the centre of the base. All its side edges are equal and all side faces are equal isosceles triangles. The height of one side triangle is the slant height (apothem) l. In a regular square pyramid with base side a and height h: l = √(h² + (a/2)²).
A tetrahedron is a triangular pyramid (4 faces). Special cases: a regular tetrahedron has 4 equilateral faces; a right-angled (trirectangular) tetrahedron has three edges at one vertex, each perpendicular to the other two, like the corner of a box.
A frustum is what is left when a plane parallel to the base cuts off the top of a pyramid. It has two parallel, similar bases and trapezium side faces.
Euler's formula: V − E + F = 2
Leonhard Euler noticed in 1750 that for every convex polyhedron, V − E + F = 2.
| Solid | V | E | F | V − E + F |
|---|---|---|---|---|
| Cube | 8 | 12 | 6 | 2 |
| Triangular prism | 6 | 9 | 5 | 2 |
| Square pyramid | 5 | 8 | 5 | 2 |
| Hexagonal prism | 12 | 18 | 8 | 2 |
Use it to find a missing number, or to test whether a solid with given V, E, F can exist. It does not work for solids with a hole through them (like a picture frame), where V − E + F = 0.
Regular polyhedra: the five Platonic solids
A polyhedron is regular when all faces are equal regular polygons and the same number of faces meet at every vertex. There are exactly five:
| Solid | Faces | At each vertex | V | E | F |
|---|---|---|---|---|---|
| Tetrahedron | triangles | 3 | 4 | 6 | 4 |
| Cube (hexahedron) | squares | 3 | 8 | 12 | 6 |
| Octahedron | triangles | 4 | 6 | 12 | 8 |
| Dodecahedron | pentagons | 3 | 20 | 30 | 12 |
| Icosahedron | triangles | 5 | 12 | 30 | 20 |
Why only five? At a vertex you need at least 3 faces, and their angles must add to less than 360°. Triangles (60°): 3, 4 or 5 work (180°, 240°, 300°); 6 makes 360°, flat. Squares (90°): only 3 (270°). Pentagons (108°): only 3 (324°). Hexagons (120°): 3 already make 360°. So only 5 cases exist.
In design: the cube and tetrahedron give strong frames; geodesic domes and many virus shells are based on the icosahedron; Platonic shapes appear in lamps, packaging and space-frame roofs.
Try it: build a cube and a tetrahedron from a net
Draw six equal squares in a cross shape on card, cut out the cross, fold along the lines and tape it: you have a cube. Count its faces, edges and vertices and check V − E + F = 2. Then draw four equilateral triangles as one big triangle split into four, fold the three corners up and make a tetrahedron. Predict its V, E, F before you count.
Key formulas and definitions
- Euler (convex polyhedron): V − E + F = 2
- Prism with n-gon base: F = n + 2, E = 3n, V = 2n
- Pyramid with n-gon base: F = n + 1, E = 2n, V = n + 1
- Cuboid diagonal: d = √(a² + b² + c²); cube: d = a√3
- Regular square pyramid: slant height l = √(h² + (a/2)²), side edge = √(h² + a²/2)
Worked examples
1. A prism has a pentagonal base. Find F, E and V and check Euler's formula.
n = 5. F = 5 + 2 = 7, E = 3 × 5 = 15, V = 2 × 5 = 10. V − E + F = 10 − 15 + 7 = 2. ✓
2. A polyhedron has 12 vertices and 30 edges. How many faces does it have?
V − E + F = 2 → 12 − 30 + F = 2 → F = 20. (It could be an icosahedron.)
3. A pyramid has 10 edges. What is its base, and how many faces does it have?
E = 2n = 10 → n = 5, so the base is a pentagon. F = n + 1 = 6.
4. A cuboid measures 3 cm × 4 cm × 12 cm. Find its space diagonal.
d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 cm.
5. A regular square pyramid has base side 6 cm and height 4 cm. Find the slant height and a side edge.
Slant height l = √(4² + 3²) = √25 = 5 cm. Side edge = √(h² + a²/2) = √(16 + 18) = √34 ≈ 5.83 cm.
6. Show that no regular polyhedron can have hexagonal faces.
Each angle of a regular hexagon is 120°. A vertex needs at least 3 faces: 3 × 120° = 360°. The angles would lie flat, but a solid corner needs a sum below 360°. So hexagons cannot form a regular polyhedron.
Common mistakes
- Counting the hidden back edges twice (or missing them). Count edges as (sum of sides of all faces) ÷ 2, because each edge is shared by two faces.
- Calling a cylinder or cone a polyhedron. They have curved surfaces, so they are not polyhedra.
- Mixing up the height of a pyramid with its slant height. Height goes straight down to the base; slant height runs down the middle of a side face.
- Thinking a frustum is cut by any plane. The cutting plane must be parallel to the base.