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3D Geometry

Shapes, Platonic solids, coordinates, and cross-sections.

3D Geometry & Polyhedra Handbook

Mathematical Principles & Theorems

Investigates three-dimensional Euclidean solids and spatial metrics. Euler's Polyhedral Formula \(V - E + F = 2\) holds for all convex polyhedra. The five regular Platonic Solids: Tetrahedron (4V, 6E, 4F), Cube (8V, 12E, 6F), Octahedron (6V, 12E, 8F), Dodecahedron (20V, 30E, 12F), and Icosahedron (12V, 30E, 20F). 3D rotation matrices \(R_x(\alpha), R_y(\beta), R_z(\gamma)\) transform vertex coordinates in \(\mathbb{R}^3\).

Operating Instructions

  • Select a 3D solid from the catalog: Platonic solids, Prism, Cylinder, Cone, or Sphere.
  • Configure edge length, radius, or height using the numeric parameter inputs.
  • Click and drag on the 3D visual stage to rotate the polyhedron in real time.
  • Inspect calculated metrics: Exact Volume, Surface Area, Vertex/Edge/Face topological counts.

3D Shape Calculator