Question

Use Euler's formula to determine the missing quantity.

Geometric Solid Number of Faces Number of Edges Number of Vertices
triangular prism 5
6
rectangular prism
12 8
pentagonal prism 7 15
hexagonal prism
18 12
octagonal prism 10
16
triangular pyramid 4 6
rectangular pyramid
8 5
pentagonal pyramid 6
6

Answer 1

Use Euler's formula V - E + F = 2.

Computed missing values:

- Triangular prism: F = 5, V = 6 → E = 6 + 5 - 2 = 9
- Rectangular prism: E = 12, V = 8 → F = 2 + 12 - 8 = 6
- Pentagonal prism: F = 7, E = 15 → V = 2 + 15 - 7 = 10
- Hexagonal prism: E = 18, V = 12 → F = 2 + 18 - 12 = 8
- Octagonal prism: F = 10, V = 16 → E = 16 + 10 - 2 = 24
- Triangular pyramid: F = 4, E = 6 → V = 2 + 6 - 4 = 4
- Rectangular pyramid: E = 8, V = 5 → F = 2 + 8 - 5 = 5
- Pentagonal pyramid: F = 6, V = 6 → E = 6 + 6 - 2 = 10

Completed table:

Geometric Solid — Faces — Edges — Vertices
- Triangular prism: 5 — 9 — 6
- Rectangular prism: 6 — 12 — 8
- Pentagonal prism: 7 — 15 — 10
- Hexagonal prism: 8 — 18 — 12
- Octagonal prism: 10 — 24 — 16
- Triangular pyramid: 4 — 6 — 4
- Rectangular pyramid: 5 — 8 — 5
- Pentagonal pyramid: 6 — 10 — 6