The Five Platonic Solids: Names, Faces, Edges & Vertices

 

Platonic Solids: Meet the Five Perfect Polyhedra.

Why are there exactly five Platonic solids—not four or six? The answer comes from a simple counting formula, the angles of regular polygons, and one important rule about what can meet at a corner. Let’s explore the five shapes and use a few quick checks to make their face, edge, and vertex counts stick.

What makes a solid Platonic?

A Platonic solid is a convex three-dimensional shape whose faces are all identical regular polygons, with the same number of faces meeting at every vertex. A regular polygon has equal sides and equal angles. “Convex” means the solid has no dents or inward caves.

The five possibilities use only equilateral triangles, squares, or regular pentagons as faces. Their names can sound unfamiliar at first, but their counts follow a pattern.

The five solids at a glance

Here, F means faces, E means edges, V means vertices (corners), and q is the number of faces meeting at each vertex.

Solid	Face shape	F	E	V	q	V − E + F
Tetrahedron	Equilateral triangle	4	6	4	3	4 − 6 + 4 = 2
Cube (hexahedron)	Square	6	12	8	3	8 − 12 + 6 = 2
Octahedron	Equilateral triangle	8	12	6	4	6 − 12 + 8 = 2
Dodecahedron	Regular pentagon	12	30	20	3	20 − 30 + 12 = 2
Icosahedron	Equilateral triangle	20	30	12	5	12 − 30 + 20 = 2

On a small phone screen, swipe horizontally inside the table to view all columns.

The key counting formula

For every convex polyhedron, Euler’s formula connects its vertices, edges, and faces:

V − E + F = 2

For example, a cube has 8 vertices, 12 edges, and 6 faces. Substituting gives 8 − 12 + 6 = 2. This is a fast way to check that a set of counts is consistent.

Why are there exactly five?

Imagine p identical regular polygon sides on each face and q faces meeting at every vertex. A regular p-sided polygon has interior angle 180° × (p − 2) ÷ p. In a convex solid, the face angles around one vertex must add to less than 360°; if they add to 360°, they lie flat instead of making a corner.

So the angle condition is:

q × 180° × (p − 2) ÷ p < 360°

Rearranging gives a compact integer rule. Because p and q are whole numbers at least 3, only five pairs satisfy it:

(p − 2) × (q − 2) < 4

The pairs are (3,3), (3,4), (3,5), (4,3), and (5,3). They give the tetrahedron, octahedron, icosahedron, cube, and dodecahedron, respectively. Each pair can form one of the five convex regular solids—no other integer pair can.

A quick angle check makes the cutoff feel concrete: a regular pentagon has 108° corners, so three meeting at a vertex make 324°, which can fold into a solid. Four would make 432°, which is too much. With regular hexagons, even three 120° corners total 360°, so they lie flat and cannot make a convex vertex.

Worked example: count a tetrahedron from its faces

A tetrahedron has 4 triangular faces, and 3 faces meet at each vertex. Count face-corners first: 4 faces × 3 corners each = 12 face-corners. Each actual vertex accounts for 3 of those corners.

V = (4 × 3) ÷ 3 = 4 vertices

Each of the 4 triangular faces has 3 sides, so counting face-sides gives 4 × 3 = 12. Every edge borders 2 faces, so divide by 2:

E = (4 × 3) ÷ 2 = 6 edges

Now check Euler’s formula:

V − E + F = 4 − 6 + 4 = 2

Memory tip: use the dual pairs

The tetrahedron has 4 faces and 4 vertices. The cube and octahedron trade face and vertex counts: (6, 8) and (8, 6). The dodecahedron and icosahedron do the same: (12, 20) and (20, 12). In each pair, the number of edges stays the same. This face–vertex swap is called duality.

To remember the names, say them in order: tetrahedron, cube, octahedron, dodecahedron, icosahedron. Then use the table to attach each face shape and count.

Common mistakes to avoid

• Mixing up faces, edges, and vertices. Faces are flat polygon regions; edges are where two faces meet; vertices are corners.
• Counting every face-side as a different edge. Each edge belongs to two faces, so divide the total face-side count by 2.
• Forgetting q. To find vertices from face-corners, divide p × F by the number q of faces meeting at each vertex.
• Assuming any regular polygon works. The angles around a convex vertex must sum to less than 360°.
• Using Euler’s formula without the right setting. V − E + F = 2 applies to convex polyhedra (and, more broadly, polyhedra topologically equivalent to a sphere).

Practice questions

1. An octahedron has 8 triangular faces, and 4 faces meet at each vertex. Find its number of vertices and edges, then check Euler’s formula.
2. A regular solid has 12 pentagonal faces, with 3 faces meeting at each vertex. Identify it, find its vertices and edges, and verify V − E + F = 2.
3. A cube has 6 square faces. Use face-corners to find its vertices and edges, given that 3 faces meet at each vertex. Then verify Euler’s formula.
4. Could 6 equilateral triangles meet at one vertex of a convex regular polyhedron? Explain using their angle sum.

On a small phone screen, swipe horizontally inside the table or an equation box to view its full contents.

Answers

1. Octahedron: Face-corners = 8 × 3 = 24. With 4 at each vertex, V = 24 ÷ 4 = 6. Edges E = 24 ÷ 2 = 12. Check: 6 − 12 + 8 = 2.
2. Dodecahedron: 12 regular pentagons identify the solid. Face-corners = 12 × 5 = 60. V = 60 ÷ 3 = 20; E = 60 ÷ 2 = 30. Check: 20 − 30 + 12 = 2.
3. Cube: Face-corners = 6 × 4 = 24. V = 24 ÷ 3 = 8; E = 24 ÷ 2 = 12. Check: 8 − 12 + 6 = 2.
4. No. An equilateral triangle has a 60° interior angle. Six around a vertex total 6 × 60° = 360°, which is flat, not a convex corner.

The takeaway

There are exactly five convex regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Remember Euler’s check, V − E + F = 2, and the angle rule: the regular face corners around a vertex must add to less than 360°.