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 for each corner angle of a regular polygon is (n - 2)(180o) / n
There are 360o around a point.
Each corner angle of a square has 90o.
Each corner angle of a regular pentagon has (5 - 2)(180o) / 5 = 108o.
Subtracting: 360o - 90o - 108o = 162o.
Since each corner angle of this regular polygon has 108o, we can find the number of sides:
(n - 2)(180o)/n = 162o
(n - 2)(180o) = 162o · n
180o·n - 360o = 162o·n
180o·n - 162o·n = 360o
18o·n = 360o
n = 20 sides