If 4.8.n is a possible vertex type for a vertex surrounded by regular polygons, what is n?
A) 6
B) 8
C) 11
D) 12

Question

Quizplus · Accepted Answer

The vertex type 4.8.n means that there are 4 faces meeting at the vertex, and one of those faces is an octagon (8 sides). The remaining n faces are regular polygons.
In order for this to be a possible vertex type, the number of sides of the regular polygons must be such that the angles around the vertex add up to exactly 360 degrees (the total angle around a point). 
For 4 faces, the sum of angle measures is 360 degrees. If we let x be the angle measure of each regular polygon face, then we have:
8 + x + x + x + x = 360
Simplifying, we get:
4x + 8 = 360
4x = 352
x = 88
Therefore, the regular polygons surrounding the octagon must each have 88 sides. The only answer choice that matches this is B.

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