Problem 4:Show that if the number of sides n of a polygon inscribed inside a circle of radius R, is very large then the area of the polygon may be approximated by the area of the circumscribed circle with radius R. (HINT: If angle x is very small and is in radians, then sin x may be approximated by x).
Solution to Problem 4:
The area of a regular polygon with n sides may be given in terms of R by
area = (1/2) n R 2 sin (2 pi / n)
If n is large, then 2 Pi /n is very small and sin (2 pi/n) may be approximated by 2 pi / n so that the area may be approximated by
area = (1/2) n R 2 (2 pi / n)
= pi R 2
which is the area of the circle.
For more on the above question, see the interactive tutorial in regular polygons.