Prove that a regular polygon with infinite sides represents a circle.Please give a mathematical proof.

Hi Speed racer,

 Mathematically, by the limit theorems, we have

  lim   magnitude of an interior angle of an n-sided regular polygon 
^n->inf 
= lim   (pi + 2×pi/n) 
  ^n->inf 
= pi

Similarly,

  lim   magnitude of an exterior angle of an n-sided regular polygon 
^n->inf 
= pi

The above calculations implies the existence of tangent which touches a point (or a vertex if we think a circle in terms of a infinite-sided regular polygon) having the interior angle equals to the exterior angle equals pi. Further evidence is the fact that the area of the infinite-sided regular polygon is in fact identical to the area of a circle.

  lim   1/2 × n×r²sin(2×pi/n) 
^n->inf 
= lim   (1/2 ×r²×2×pi/n)×(sin(2×pi/n)/(2×pi/n)) 
 ^n->inf 
(Recall lim   sin(theta)/theta = 1) 
        ^theta->0 
= pi×r²

Approved