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A Regular Polygon with 137 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 137 sides where the length of each side is 5 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 137 sides. What are the interior angles and exterior angles? 
If the length of each side is 5 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 137

Perimeter of a polygon with 137 sides = (side length) x 137 = 685 units

Area of a polygon with 137 sides = (n x Sidecot (Π/n))/4 = (n x 5cot (Π/137))/4 = 37333.19 square units

Sum of the interior angles of a polygon with 137 sides = 
 (n-2) x 180 degrees =  (137-2) x 180 degrees = 24300 degrees

Interior Angle of a polygon with 137 sides = (n-2) x 180/n degrees = (137-2) x 180/137 degrees = 177.37 degrees 

Exterior angle of a polygon with 137 sides = 180 - Interior Angle = 180 - 177.37 = 2.62 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/137) = 5 x cot 1 = 218 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/137) = 5 x cosec 1 degrees = 218.06 units

Symmetry Group = D137  137 rotational symmetries and 137 reflection symmetries. The "D" stands for di-hedral. 




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