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A Regular Polygon with 128 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 128 sides where the length of each side is 2 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 128 sides. What are the interior angles and exterior angles? 
If the length of each side is 2 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 128

Perimeter of a polygon with 128 sides = (side length) x 128 = 256 units

Area of a polygon with 128 sides = (n x Sidecot (Π/n))/4 = (n x 2cot (Π/128))/4 = 5214.14 square units

Sum of the interior angles of a polygon with 128 sides = 
 (n-2) x 180 degrees =  (128-2) x 180 degrees = 22680 degrees

Interior Angle of a polygon with 128 sides = (n-2) x 180/n degrees = (128-2) x 180/128 degrees = 177.18 degrees 

Exterior angle of a polygon with 128 sides = 180 - Interior Angle = 180 - 177.18 = 2.81 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/128) = 2 x cot 1 = 81.47 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/128) = 2 x cosec 1 degrees = 81.49 units

Symmetry Group = D128  128 rotational symmetries and 128 reflection symmetries. The "D" stands for di-hedral. 




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