A semiregular tiling has an equilateral triangle, a regular hexagon and two regular p-gons at each vertex. What is p?
A) 3
B) 4
C) 5
D) 6

Question

Quizplus · Accepted Answer

One way to think about this is to look at the angles around a vertex. A full rotation around a point is $360^\circ$, and at each vertex, there are two equilateral triangles and one regular hexagon meeting. The sum of the angles in an equilateral triangle is $60^\circ$, so the sum of the angles in two equilateral triangles is $120^\circ$. The sum of the angles in a regular hexagon is $720^\circ$, so the sum of the angles in the three polygons meeting at a vertex is $840^\circ$. Therefore, we have:

$$840^\circ = 60^\circ + 60^\circ + \frac{p}{2}360^\circ$$

Solving for $p$, we get $p = \boxed{4}$.

A Semiregular Tiling Has an Equilateral Triangle, a Regular Hexagon