For any sided regular polygon inscribed in a circle of radius , our objective is to calculate the perimeter of the polygon.
The figure represents the given situation. is the center of the circle and , are edges of a regular polygon. and are perpendiculars drawn on line segments and respectively. Since perpendiculars from center bisect the chord, thus we have,
Therfore by RHS Criteration of congruency, .
Thus by CPCT, we have
For any regular polygon of sides, each angle is given by .
Therefore, . Thus .
Since it is a regular polygon,
But as approaches infinity the polygon tends to coincide the circle in which it is inscribed. Thus in that case the perimeter of the polygon becomes equal to the circumference of the circle in which it is inscribed.
Also As approaches infinity,
becomes approximately .
Also comparing the formula we obtained in equation 1 , to the circumference of circle,
From (2) and (3),
Dividing by on both sides we get,
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