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A Regular Polygon with 51 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 51 sides where the length of each side is 10 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 51 sides. What are the interior angles and exterior angles? 
If the length of each side is 10 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 51

Perimeter of a polygon with 51 sides = (side length) x 51 = 510 units

Area of a polygon with 51 sides = (n x Sidecot (Π/n))/4 = (n x 10cot (Π/51))/4 = 20671.91 square units

Sum of the interior angles of a polygon with 51 sides = 
 (n-2) x 180 degrees =  (51-2) x 180 degrees = 8820 degrees

Interior Angle of a polygon with 51 sides = (n-2) x 180/n degrees = (51-2) x 180/51 degrees = 172.94 degrees 

Exterior angle of a polygon with 51 sides = 180 - Interior Angle = 180 - 172.94 = 7.05 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/51) = 10 x cot 3 = 162.13 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/51) = 10 x cosec 3 degrees = 162.44 units

Symmetry Group = D51  51 rotational symmetries and 51 reflection symmetries. The "D" stands for di-hedral. 




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