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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 15, 2020
edited by Guest  May 15, 2020
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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

 May 15, 2020

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