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Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point. An equilateral triangle, a regular octagon, and a regular n-gon, all with the same side length, also completely surround a point. Find n.

 Jan 17, 2020
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The formula to find the number of degrees in each vertex angle of a regular polygon is:  Degrees = (n - 2) · 180 / n

 

Each angle of an equilateral triangle is 60°.

In a regular octagon, n = 8  and  Degrees = (n - 2) · 180 / n     --->      Degrees  =  (8 - 2) · 180 / 8     --->    Degrees  =  135°.

 

Since there are  360°  around a point, the number of degrees in each vertex of the regular n-gon is: 

                             360° - 60° - 135°  =  165°.

 

To find the value of n:  165°  =  (n - 2) · 180 / n   

 --->                          165 · n  =  (n - 2) · 180

 --->                             165n  =  180n - 360

 --->                              -15n  =  -360

 --->                                    n =  24

 Jan 17, 2020

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