There is a regularity in the numbers of faces, edges, and vertices of polyhedra. Remember Euler's polyhedron formula.
Euler's Polyhedron Formula
(number of faces) + (number of vertices) - (number of edges) = 2
📘 Example ①
Find the numbers of faces, edges, and vertices of a triangular prism, and verify Euler's formula.
Answer: 5 faces (2 triangles + 3 rectangles), 9 edges (3 base edges×2 + 3 lateral edges), and 6 vertices (3 vertices on each base×2).
Verification: 5 + 6 - 9 = 2 ✓
Find the numbers of faces, edges, and vertices of a triangular prism, and verify Euler's formula.
Answer: 5 faces (2 triangles + 3 rectangles), 9 edges (3 base edges×2 + 3 lateral edges), and 6 vertices (3 vertices on each base×2).
Verification: 5 + 6 - 9 = 2 ✓
📘 Example ②
Find the numbers of faces, edges, and vertices of a square pyramid.
Answer: 5 faces (1 square + 4 triangles), 8 edges (4 base edges + 4 lateral edges), and 5 vertices (4 base vertices + 1 vertex).
Verification: 5 + 5 - 8 = 2 ✓
Find the numbers of faces, edges, and vertices of a square pyramid.
Answer: 5 faces (1 square + 4 triangles), 8 edges (4 base edges + 4 lateral edges), and 5 vertices (4 base vertices + 1 vertex).
Verification: 5 + 5 - 8 = 2 ✓
💡 Key Points
- Euler's formula: faces + vertices - edges = 2 (true for all polyhedra)
- n-gonal prism: n+2 faces, 3n edges, 2n vertices
- n-gonal pyramid: n+1 faces, 2n edges, n+1 vertices
Practice Problems
- Find the numbers of faces, edges, and vertices of a hexagonal prism, and verify them using Euler's formula.
- Find the numbers of faces, edges, and vertices of a pentagonal pyramid.
Answers and Explanations
- Answer: 8 faces, 18 edges, and 12 vertices. Verification: 8+12-18=2 ✓.
- Answer: 6 faces, 10 edges, and 6 vertices. Verification: 6+6-10=2 ✓.