What kind of regular polygons should placed in this area?

Question

A regular polygon has to be filled in at the vertex with angle marked $ (z \, 150^{\circ}) $ to form a 3D tessellation.

What kind of regular polygon should be placed in this area to complete the tesselation?

Answer 1

A regular $n$-gon has interior angle $\pi-2\pi/n$.

Here, to fit exactly, you want to get an angle of 150°, or $5\pi/6$.

Solve $\dfrac{n-2}{n}\pi=\dfrac56\pi$, and you get $n=12$.

Answer 2

An interior angle of $150°$ implies an exterior angle of $180-150 = 30°$, so the regular polygon that would fill the $z$ angle on its own would have $360 / 30 = 12$ sides, since the exterior angles of every convex polygon sum to $360°$.

(If you are allowed two regular polygons to fill that angle, they would be a square and an equilateral triangle.)

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Diagram for clarity - the exterior angle measures the change in direction at a corner for something traversing the perimeter of the polygon