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A Regular Polygon with 84 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 84 sides where the length of each side is 4 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 84 sides. What are the interior angles and exterior angles? 
If the length of each side is 4 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 84

Perimeter of a polygon with 84 sides = (side length) x 84 = 336 units

Area of a polygon with 84 sides = (n x Sidecot (Π/n))/4 = (n x 4cot (Π/84))/4 = 8979.78 square units

Sum of the interior angles of a polygon with 84 sides = 
 (n-2) x 180 degrees =  (84-2) x 180 degrees = 14760 degrees

Interior Angle of a polygon with 84 sides = (n-2) x 180/n degrees = (84-2) x 180/84 degrees = 175.71 degrees 

Exterior angle of a polygon with 84 sides = 180 - Interior Angle = 180 - 175.71 = 4.28 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/84) = 4 x cot 2 = 106.9 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/84) = 4 x cosec 2 degrees = 106.97 units

Symmetry Group = D84  84 rotational symmetries and 84 reflection symmetries. The "D" stands for di-hedral. 




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