Polyhedrons & Euler's Formula
A polyhedron is a 3D solid bounded by flat faces. Euler's Formula: connects vertices, edges, and faces for any convex polyhedron.
Introduction
Past Knowledge
Polygon properties (9.1). Points, edges, faces in 2D.
Today's Goal
Classify polyhedrons and verify Euler's formula V − E + F = 2.
Future Success
Cross-sections (12.1.2), surface area (12.2), volume (12.3).
Key Concepts
Euler's Formula
For any convex polyhedron: Vertices − Edges + Faces = 2
The Five Platonic Solids
| Solid | V | E | F | Faces |
|---|---|---|---|---|
| Tetrahedron | 4 | 6 | 4 | Triangles |
| Cube | 8 | 12 | 6 | Squares |
| Octahedron | 6 | 12 | 8 | Triangles |
| Dodecahedron | 20 | 30 | 12 | Pentagons |
| Icosahedron | 12 | 30 | 20 | Triangles |
Euler's Formula Verification
Verify for a cube: V = 8, E = 12, F = 6.
Verify for a triangular prism: V = 6, E = 9, F = 5.
∎ Works for every convex polyhedron without holes.
Worked Examples
Find Missing Value
V = 10, F = 7. Find E.
10 − E + 7 = 2 → E = 15
E = 15
Common Pitfalls
Polyhedron ≠ All 3D Shapes
Cylinders, cones, and spheres are NOT polyhedrons — they have curved surfaces. Euler's formula applies to flat-faced solids only.
Real-Life Applications
Dice & Game Design
D4, D6, D8, D12, and D20 dice correspond to the 5 Platonic solids. Euler's formula confirms each is a valid polyhedron.
Geodesic Domes
Buckminster Fuller's geodesic domes are based on icosahedrons. Euler's formula helps verify structural integrity.
Practice Quiz
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