Question

Euler’s formula defines a distinct relationship between the number of faces, vertices, and edges of a solid. Use Euler’s formula to determine the missing quantity in each row of the table.

Geometric Solid Number of Faces Number of Edges Number of Vertices
cube
12 8
triangular prism 5
6
pentagonal pyramid 6 10

Answer 1

Use Euler’s formula for convex polyhedra: V − E + F = 2.

1. Cube: E = 12, V = 8.
F = 2 + E − V = 2 + 12 − 8 = 6.

2. Triangular prism: F = 5, V = 6.
E = V + F − 2 = 6 + 5 − 2 = 9.

3. Pentagonal pyramid: F = 6, E = 10.
V = 2 + E − F = 2 + 10 − 6 = 6.

Completed table:
- Cube: Faces = 6, Edges = 12, Vertices = 8
- Triangular prism: Faces = 5, Edges = 9, Vertices = 6
- Pentagonal pyramid: Faces = 6, Edges = 10, Vertices = 6