3D Geometry

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

3D Shape Calculator

Platonic Solids

Regular, convex polyhedra with congruent faces of regular polygons.

3D Coordinates

Point 1 (Cartesian)

x

y

z

Point 2 (Distance/Midpoint)

P₂ x

P₂ y

P₂ z

Conic Sections

Cross-sections of a cone.

Transformations & Properties

Cylindrical (r, φ, z)

Spherical (ρ, θ, φ)

Properties

P₁ to P₂ Distance

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.