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.

Logic Apr 10, 2019

#1**+1 **

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)(180^{o})

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

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

Add that to the 60^{o} of the triangle angle touching the octagon angle

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

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

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

Guest Apr 10, 2019