Question

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

Explanation:

Step1: Recall Euler's Formula

Euler's Formula: $V - E + F = 2$

Step2: Calculate total edge incidences

Octagon has 8 edges, 8 triangles have $8×3=24$ edges. Each edge shared by 2 faces: $E=\frac{8+24}{2}=16$

Step3: Find number of vertices

$F=9$, so $V=2 + E - F=2+16-9=10$

Answer:

10