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A Regular Polygon with 27 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 27 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 27 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 = 27

Perimeter of a polygon with 27 sides = (side length) x 27 = 135 units

Area of a polygon with 27 sides = (n x Sidecot (Π/n))/4 = (n x 5cot (Π/27))/4 = 1443.74 square units

Sum of the interior angles of a polygon with 27 sides = 
 (n-2) x 180 degrees =  (27-2) x 180 degrees = 4500 degrees

Interior Angle of a polygon with 27 sides = (n-2) x 180/n degrees = (27-2) x 180/27 degrees = 166.66 degrees 

Exterior angle of a polygon with 27 sides = 180 - Interior Angle = 180 - 166.66 = 13.33 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/27) = 5 x cot 6 = 42.77 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/27) = 5 x cosec 6 degrees = 43.06 units

Symmetry Group = D27  27 rotational symmetries and 27 reflection symmetries. The "D" stands for di-hedral. 




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