Kemal G. answered • 03/16/17

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Hi Jess,

This is a nice geometry question making use of circle and polygon properties.

First and foremost, we are given that the decagon (i.e. 10 sides) is regular so we know then that all interior angles and sides are equal.

Also,

**the radius of a polygon means the distance from the center of a regular polygon to any vertex.**You will notice that when we connect the center to the vertices of the polygon, we get 10 partitions (not surprising since it is a decagon!). This means that the angle of each triangle around the center is 360/10=36 degrees.

At this point, I hope you can see that we have ten congruent triangles in the polygon whose smallest angles are 36 degrees. Moreover, they are isosceles since their two sides are the radii of the circle and the unknown sides are the sides of the polygon!

Now, we can simply use the law of cosines to find the side length of the polygon

if we call the side length x

x^2 = 14^2 + 14^2 -2*14*14*cos(36deg)

x = 8.65' (to the nearest hundredth)

Then the perimeter is

**8.65*10 = 86.5'**

Or if you just want to memorize the formula and use it, then here it is

r = s/2sin(180/n) where r is the radius, s is the length of any side, n is the number of sides and the angle measure is in degrees.