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Geometry › Solid geometry
Solid geometry
Shapes that take up space — vertices, edges, faces. Plus Euler's tiny formula that connects them all.
3D solid gallery
Tap a solid. See its V, E, F — and check Euler's formula.
Selected
Cube
V
8
E
12
F
6
6 square faces. Also called a hexahedron. The dice of dice.
Euler's formula
V − E + F = 8 − 12 + 6 = 2
Polyhedra and non-polyhedra
A polyhedron is a 3D solid bounded entirely by flat polygon faces. Cubes, pyramids, prisms — all polyhedra. Spheres, cylinders, cones are non-polyhedra because they include curved surfaces.
Three things to count
- Vertices (V) — corner points.
- Edges (E) — straight line segments where two faces meet.
- Faces (F) — flat surfaces.
Euler's formula
For any convex polyhedron:
A cube: 8 − 12 + 6 = 2. A tetrahedron: 4 − 6 + 4 = 2. A dodecahedron: 20 − 30 + 12 = 2.
V − E + F = 2A cube: 8 − 12 + 6 = 2. A tetrahedron: 4 − 6 + 4 = 2. A dodecahedron: 20 − 30 + 12 = 2.
The Platonic solids
Only five convex polyhedra have all faces identical regular polygons with the same number meeting at each vertex:
- Tetrahedron — 4 triangular faces.
- Cube (hexahedron) — 6 square faces.
- Octahedron — 8 triangular faces.
- Dodecahedron — 12 pentagonal faces.
- Icosahedron — 20 triangular faces.
Plato thought the universe was built from them. Modern dice still use them: d4, d6, d8, d12, d20.
Prisms and pyramids
- A prism has two identical parallel polygon ends connected by rectangles.
- A pyramid has one polygon base and triangular faces meeting at an apex.
Curved solids
- Sphere — every point equidistant from center.
- Cylinder — two circular ends joined by a curved surface.
- Cone — circular base, single apex.
- Torus — donut shape: a circle swept around an axis.