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A Regular Polygon with 21 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 21 sides where the length of each side is 6 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 21 sides. What are the interior angles and exterior angles? 
If the length of each side is 6 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 21

Perimeter of a polygon with 21 sides = (side length) x 21 = 126 units

Area of a polygon with 21 sides = (n x Sidecot (Π/n))/4 = (n x 6cot (Π/21))/4 = 1253.93 square units

Sum of the interior angles of a polygon with 21 sides = 
 (n-2) x 180 degrees =  (21-2) x 180 degrees = 3420 degrees

Interior Angle of a polygon with 21 sides = (n-2) x 180/n degrees = (21-2) x 180/21 degrees = 162.85 degrees 

Exterior angle of a polygon with 21 sides = 180 - Interior Angle = 180 - 162.85 = 17.14 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/21) = 6 x cot 8 = 39.8 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/21) = 6 x cosec 8 degrees = 40.25 units

Symmetry Group = D21  21 rotational symmetries and 21 reflection symmetries. The "D" stands for di-hedral. 




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