Britannica Sources used in this answer

What kinds of solid figures are studied in Euclidean geometry?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclidean geometry, solid geometry focuses on three-dimensional figures, which are often studied as analogies to two-dimensional shapes. Key solid figures include:1

Polyhedra: These are three-dimensional objects composed of a finite number of polygonal surfaces (faces) that meet at edges and vertices.2
Platonic solids: These are the five regular polyhedra whose faces are congruent regular polygons meeting at congruent polyhedral angles. They include:34
- Tetrahedron: 4 triangular faces.34
- Cube: 6 square faces.34
- Octahedron: 8 equilateral triangular faces.34
- Dodecahedron: 12 pentagonal faces.34
- Icosahedron: 20 equilateral triangular faces.34
Sphere: Defined as the set of all points in three-dimensional space at a constant distance (the radius) from a central point.5

Beyond three dimensions, Euclidean geometry also generalizes these concepts into higher-dimensional spaces through the study of polytopes.3