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 $n$.

Guest May 11, 2020

#2**+1 **

The formula to find the number of degrees in each vertex angle of a regular polygon:

degrees = (n - 2)(180)/n

There are 360^{o} around each point.

Each vertex angle of a square has 90^{o}.

Each vertex angle of a regular pentagon has 108^{o}. [ (5-2)(180)/5 = 108 ]

There are 360^{o} - 90^{o} - 108^{o} = 162^{o} 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.

geno3141 May 11, 2020