Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.
A square, a regular pentagon, and a regular $n$-gon, all with the same side length, also completely surround a point. Find $n$.
The formula to find the number of degrees in each vertex angle of a regular polygon:
degrees = (n - 2)(180)/n
There are 360o around each point.
Each vertex angle of a square has 90o.
Each vertex angle of a regular pentagon has 108o. [ (5-2)(180)/5 = 108 ]
There are 360o - 90o - 108o = 162o remaining.
Using the formula: (n - 2)(180)/n = 162
(n - 2)(180) = 162n
180n - 360 = 162n
-360 = -18n
20 = n
It's a 20-sided regular polygon.