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A Regular Polygon with 45 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 45 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 45 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 = 45

Perimeter of a polygon with 45 sides = (side length) x 45 = 225 units

Area of a polygon with 45 sides = (n x Sidecot (Π/n))/4 = (n x 5cot (Π/45))/4 = 4022.06 square units

Sum of the interior angles of a polygon with 45 sides = 
 (n-2) x 180 degrees =  (45-2) x 180 degrees = 7740 degrees

Interior Angle of a polygon with 45 sides = (n-2) x 180/n degrees = (45-2) x 180/45 degrees = 172 degrees 

Exterior angle of a polygon with 45 sides = 180 - Interior Angle = 180 - 172 = 8 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/45) = 5 x cot 4 = 71.5 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/45) = 5 x cosec 4 degrees = 71.67 units

Symmetry Group = D45  45 rotational symmetries and 45 reflection symmetries. The "D" stands for di-hedral. 




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