The Learning Point‎ > ‎Mathematics‎ > ‎Polygons‎ > ‎

A Regular Polygon with 85 sides: Area, Perimeter, Interior and Exterior Angles, Inradius, Circumradius



A regular polygon with 85 sides where the length of each side is 3 units. The units may either be inches or cm or km or miles: any unit of length.

You are given a Regular Polygon with 85 sides. What are the interior angles and exterior angles? 
If the length of each side is 3 units, what is the perimeter, area, circum-radius and in-radius of the polygon?

n = 85

Perimeter of a polygon with 85 sides = (side length) x 85 = 255 units

Area of a polygon with 85 sides = (n x Sidecot (Π/n))/4 = (n x 3cot (Π/85))/4 = 5172.16 square units

Sum of the interior angles of a polygon with 85 sides = 
 (n-2) x 180 degrees =  (85-2) x 180 degrees = 14940 degrees

Interior Angle of a polygon with 85 sides = (n-2) x 180/n degrees = (85-2) x 180/85 degrees = 175.76 degrees 

Exterior angle of a polygon with 85 sides = 180 - Interior Angle = 180 - 175.76 = 4.23 degrees

Inradius = Radius of In-circle = (side length) x cot (Π/85) = 3 x cot 2 = 81.13 units

Circumradius = Radius of Circum-circle = (side length) x cosec (Π/85) = 3 x cosec 2 degrees = 81.18 units

Symmetry Group = D85  85 rotational symmetries and 85 reflection symmetries. The "D" stands for di-hedral. 




Comments