At the vertex of a tiling of the plane there are two regular pentagons and one regular p-gon at each vertex. What is p?
A) 8
B) 9
C) 10
D) 12

Question

Quizplus · Accepted Answer

At any given vertex, the sum of the interior angles must be equal to $360^\circ$. Therefore, we have:

$2(	ext{interior angle of a pentagon})+1(	ext{interior angle of a p-gon})=360^\circ$

Using the formula for the interior angle of a regular $n$-gon $\frac{(n-2)180^\circ}{n}$, we can write:

$2(\frac{(5-2)180^\circ}{5})+1(\frac{(p-2)180^\circ}{p})=360^\circ$

Simplifying this equation gives:

$2160-720p+180p-360p=0$

$360=540p$

$p=\boxed{10}$

At the Vertex of a Tiling of the Plane There