Polyhedra

Japanese school year: Math A

What you learn

Explore the geometric properties and symmetries of the five Platonic solids, along with Euler's polyhedron formula relating vertices, edges, and faces. This topic builds on basic spatial concepts and regular polygons. It is applied in analyzing molecular crystal structures, 3D computer graphics, and architectural design.

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Key points

Known as Euler's formula, this connects vertices $v$ , edges $e$ , and faces $f$ for any convex polyhedron. The equation $v - e + f = 2$ always holds regardless of the shape.

v - e + f = 2

This relates edges $e$ , faces $f$ , and vertices $v$ of a regular polyhedron. When each face is a regular $n$ -gon and $k$ edges meet at each vertex, sharing edges yields $2e = nf = kv$ .

2e = nf = kv

There exist only five regular polyhedra whose faces are identical regular polygons: the regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron, and regular icosahedron.

Choose a set to practice.

Practice 1Coming soon