Question

1. use eulers formula to find the number of vertices in a polyhedron with nine faces: one octagon and eight triangles.

Explanation:

Step1: List known values

Faces ($F$) = 9 (1 octagon + 8 triangles), Euler's formula: $V - E + F = 2$

Step2: Calculate edges ($E$)

Each triangle has 3 edges, octagon has 8 edges; each edge shared by 2 faces: $E = \frac{(8×3 + 8)}{2} = \frac{32}{2} = 16$

Step3: Solve for vertices ($V$)

Rearrange formula: $V = E - F + 2 = 16 - 9 + 2 = 9$

Answer:

9