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Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.



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

 Apr 10, 2019
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Draw the octagon with a triangle stuck to one of the sides.

 

The sum of all the angles in a polygon with x sides is (x -2)(180o)

 

The interior angles of the octagon are (8 - 2)(180o) = 1080o

So one angle of the octagon is 1080o/8 = 135o

 

Add that to the 60o of the triangle angle touching the octagon angle

and you get a total of 195o which makes the angle on the other side

which is an interior angle of our n-gon be 360o - 195o = 165o 

 

I'm going to drop the degrees sign for convenience

 

Since n is the number of sides of our n-gon,

and there are the same number of angles as there are sides,

the total of the interior angles of our n-gon is (165)(n)  

 

But the total is also (n - 2)(180) so set them equal

 

165n = 180n - 360

165n -180n = -360

multiply both sides by -1

180n -165n = 360

15n = 360

n = 24

 Apr 10, 2019

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