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A Regular Polygon with 54 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 54 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 54 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 = 54

Perimeter of a polygon with 54 sides = (side length) x 54 = 486 units

Area of a polygon with 54 sides = (n x Sidecot (Π/n))/4 = (n x 9cot (Π/54))/4 = 18774.66 square units

Sum of the interior angles of a polygon with 54 sides = 
 (n-2) x 180 degrees =  (54-2) x 180 degrees = 9360 degrees

Interior Angle of a polygon with 54 sides = (n-2) x 180/n degrees = (54-2) x 180/54 degrees = 173.33 degrees 

Exterior angle of a polygon with 54 sides = 180 - Interior Angle = 180 - 173.33 = 6.66 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/54) = 9 x cot 3 = 154.52 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/54) = 9 x cosec 3 degrees = 154.78 units

Symmetry Group = D54  54 rotational symmetries and 54 reflection symmetries. The "D" stands for di-hedral. 




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