The Ultimate Guide to Geometry
A complete guide to Euclidean Geometry — from Euclid's axioms through plane geometry, solid geometry, Platonic Solids, and beyond.
Welcome to the Guide
This is a complete online guide to Euclidean geometry. It begins where Euclid began — with definitions, axioms, and postulates — and proceeds through plane geometry, coordinate geometry, and the rich world of three-dimensional solids, culminating in the Platonic Solids, their duals, the Archimedean and Catalan families, and compound polyhedra.
Follow the chapters in order for a systematic course, or jump to any topic that interests you. Each chapter builds on what came before, but is also written to be accessible as a standalone reference.
What is Geometry?
Geometry is the study of space and form: the shapes, sizes, positions, and relationships of points, lines, surfaces, and solids. The word comes from the Greek geo (earth) and metria (measurement) — its origins are practical, rooted in land surveying and architecture. But geometry quickly became something more: a deductive system in which everything follows from a small number of clearly stated assumptions.
This deductive structure — definitions, axioms, postulates, propositions — was set out by Euclid of Alexandria around 300 BC in his Elements, one of the most influential works in the history of mathematics. Euclid's method of proof from first principles became the model for all rigorous mathematics that followed.
Begin your geometry journey with Euclid's axioms, definitions, and the foundational logic of geometric proof that underpins all of mathematics.
Explore Part 1 →Explore the geometry of flat surfaces — triangles, polygons, circles, transformations, and the elegant relationships between 2D shapes.
- Triangles
- Quadrilaterals
- Regular Polygons and Areas
- The Circle
- Similarity and Proportion
- Trigonometry
- Euclidean Constructions
- Tessellations and Tilings
- Loci
Bridge geometry and algebra — place shapes on the coordinate plane, work with equations of lines and curves, and discover analytic geometry.
Explore Part 3 →Enter the third dimension — explore prisms, pyramids, the sphere, the Platonic and Archimedean solids, and the elegant world of three-dimensional geometry.
- Lines, Planes and Solid Angles
- Prisms and Pyramids
- Cylinders, Cones and the Sphere
- Platonic Solids
- Dual Polyhedra
- Compound Polyhedra
- Archimedean Solids
- Catalan Solids
Step beyond three dimensions into the geometry of the torus, the hypercube, and higher-dimensional polytopes revealing atomic structure.
Explore Part 5 →