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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 $n$.

 May 11, 2020
 #1
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 May 11, 2020
edited by Dragan  May 16, 2020
edited by Dragan  May 16, 2020
 #2
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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.

 May 11, 2020
 #3
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That answer is wrong.

Guest May 12, 2020
 #4
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It is correct!!! smiley

Dragan  May 16, 2020

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