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A Regular Polygon with 31 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 31 sides where the length of each side is 9 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 31 sides. What are the interior angles and exterior angles? 
If the length of each side is 9 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 31

Perimeter of a polygon with 31 sides = (side length) x 31 = 279 units

Area of a polygon with 31 sides = (n x Sidecot (Π/n))/4 = (n x 9cot (Π/31))/4 = 6173.16 square units

Sum of the interior angles of a polygon with 31 sides = 
 (n-2) x 180 degrees =  (31-2) x 180 degrees = 5220 degrees

Interior Angle of a polygon with 31 sides = (n-2) x 180/n degrees = (31-2) x 180/31 degrees = 168.38 degrees 

Exterior angle of a polygon with 31 sides = 180 - Interior Angle = 180 - 168.38 = 11.61 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/31) = 9 x cot 5 = 88.5 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/31) = 9 x cosec 5 degrees = 88.96 units

Symmetry Group = D31  31 rotational symmetries and 31 reflection symmetries. The "D" stands for di-hedral. 




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