Regular Polygon

   
Number of sides, all equal length a: n
Number of interior angles, all equal measure beta: n
Central angle subtending one side: alpha
 
Perimeter: P
Area: K
 
Radius of circumscribed circle: R
Radius of inscribed circle: r
 
    beta = p(n-2)/n radians = 180o(n-2)/n
alpha = 2 p/n radians = 360o/n
alpha + beta = p radians = 180o
 
P = na = 2nR sin(alpha/2)
K = na2 cot(alpha/2)/4
    = nR2 sin(alpha)/2
    = nr2 tan(alpha/2)
    = na sqrt(4R2-a2)/4
  
R = a csc(alpha/2)/2
r = a cot(alpha/2)/2
 
a = 2r tan(alpha/2) = 2R sin(alpha/2)
 

 

 

 

   Special Cases of the Regular Polygon

 (equilateral triangle) n = 3
 (square)                  n = 4
 (regular pentagon)   n = 5
 (regular hexagon)    n = 6
 (regular octagon)     n = 7
       The Regular Pentagon    regular pentagon  

      Number of sides n = 5
      Internal angles beta = 3p/5 radians = 108 degrees
      Central angles alpha = 2p/5 radians = 72 degrees

      Perimeter P = 5a = 5 R sqrt[10-2 sqrt(5)] / 2
      Area K = 5a2 sqrt[1+2/sqrt(5)] / 4
                  = 5R2 sqrt[10+2 sqrt(5)]/8
                  = 5r2 sqrt[5-2 sqrt(5)]
                  = 5a sqrt(4R2-a2) / 4

      Circumradius R = a sqrt[2+2/sqrt(5)] / 2
      Apothem r = a sqrt[1+2/sqrt(5)] / 2 = R[1+sqrt(5)] / 4

      Side a = 2r sqrt[5-2 sqrt(5)] = R sqrt[10-2 sqrt(5)] / 2

       The Regular Hexagon    regular hexagon

      Number of sides n = 6
      Internal angles beta = 2/3 radians = 120 degrees
      Central angles alpha = /3 radians = 60 degrees

      Perimeter P = 6a = 6R
      Area K = 3a2 sqrt(3) / 2
                  = 3R2 sqrt(3) / 2
                  = 2r2 sqrt(3)
                  = 3a sqrt(4R2-a2) / 2

      Circumradius R = a
      Apothem r = a sqrt(3) / 2 = R sqrt(3) / 2

      Side a = 2r / sqrt(3) = R

       The Regular Octagon    regular octagon

      Number of sides n = 8
      Internal angles beta = 3/4 radians = 135 degrees
      Central angles alpha = /4 radains = 45 degrees

      Perimeter P = 8a = 8R sqrt[2-sqrt(2)]
      Area K = 2a2[sqrt(2)+1]
                  = 2R2 sqrt(2)
                  = 8r2[sqrt(2)-1]
                  = 2a sqrt(4R2-a2)

      Circumradius R = a sqrt[sqrt(2) + 1]
      Apothem r = a [sqrt(2) + 1] / 2 = R sqrt[ 2 + sqrt(2) ] / 2

      Side a = 2r [sqrt(2) -1] = R sqrt[ 2 - sqrt(2)]