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A Regular Polygon with 109 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 109 sides where the length of each side is 7 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 109 sides. What are the interior angles and exterior angles? 
If the length of each side is 7 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 109

Perimeter of a polygon with 109 sides = (side length) x 109 = 763 units

Area of a polygon with 109 sides = (n x Sidecot (Π/n))/4 = (n x 7cot (Π/109))/4 = 46314.7 square units

Sum of the interior angles of a polygon with 109 sides = 
 (n-2) x 180 degrees =  (109-2) x 180 degrees = 19260 degrees

Interior Angle of a polygon with 109 sides = (n-2) x 180/n degrees = (109-2) x 180/109 degrees = 176.69 degrees 

Exterior angle of a polygon with 109 sides = 180 - Interior Angle = 180 - 176.69 = 3.3 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/109) = 7 x cot 1 = 242.8 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/109) = 7 x cosec 1 degrees = 242.9 units

Symmetry Group = D109  109 rotational symmetries and 109 reflection symmetries. The "D" stands for di-hedral. 




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