Question

If the exterior angle of a regular polygon is 45 degrees, then what is the no. of sides in the polygon and the no. of diagonals?
I need this answer with the steps and formulas used

Answer 1

sum of exterior angles = 360 degrees

no of sides(n) = sum of exterior angles of a regular polygon / given exterior angle of the regular polygon
⇒360/45
⇒8 sides.
no of diagonals= n(n-3)/2⇒8(8-3)/2⇒8(5)/2⇒40/2⇒20 diagonals
Therefore, number of sides are 8 and number of diagonals are 20

Answer 2

Please see diagram.
 
   A regular polygon has n sides and n vertices. The lines joining the vertices and the center O of polygon create n isosceles triangles. The side of the polygon becomes base of these triangles. The angle at the center in the triangle is Ф = 360°/n.
   
     As the two angles at the base are A/2 = (180° - Ф )/2.
     The interior angle at a vertex is A = 180°- Ф.
 
         So the exterior angle  is = Ф = 360°/n = 2π/n
              So if 360°/n = 45°,           n  = 360°/45 = 8
 
   It is a regular octagon with 8 sides.
   
   The number of line connecting each vertex to another is :  ⁸C₂ = 8 * 7 /2  = 28.

      Of these, there are 8 sides among adjacent vertices.
    
        The remaining are the diagonals and are 28 – 8 = 20.
 
=========================
          n >= 3
 
   Formula for number of diagonals =  n(n-1)/2  - n = n(n-3)/2.
 
          Sum total of all exterior angles = 360° = 2 π  for any regular  polygon.
 
           One exterior angle = 360° / n = 2π/n
            one interior angle = 180° – 360°/n = 180° (n-2)/n  =  (n-2)π/n
           Angle made by a side at the cente = 360°/n  = 2π/n
           Sum total of all interior angles = n * 180° (n-2)/n  = 180° (n-2)   = (n-2)π