Sn is a regular n-gon inscribed in a circle of radius 1. (A regular n-gon is a polygon with n sides and where each of the sides has the same length and the angles around the n-gon are all equal. For example, a square is a regular 4-gon, and an equilateral triangle is a regular 3-gon.)
1. The goal of this problem is to compute the perimeter pn of the regular n-gon Sn and then try to compute lim n→∞ pn. 1. (1 point) Carefully draw a diagram showing a regular 6-gon (or hexagon) inscribed in a circle of radius 1. Use geometric reasoning to compute the perimeter p6 of this hexagon.
2. Draw a radius from the center of the circle to each vertex of Sn. This divides Sn into n congruent isosceles triangles. What is the measure of the angle θ (in radians) at the center of the circle for each of these triangles?
3. Carefully explain why the length of the base of each of these isosceles triangles is equal to 2sin(π n). (From the center of the circle, draw a perpendicular line to the base of the isosceles triangle, and then express the length of the base in terms of an appropriate trigonometric function and angle.)
4. Find a formula for the perimeter pn using the answer in (3). Verify that for n = 6, your formula agrees with your answer in (1).
5. Try to ?nd lim n→∞ pn. You could use your calculator to evaluate pn for larger and larger values of n. Can you guess the correct limit by using geometric reasoning and by considering how the picture changes as n becomes arbitrarily large? Is your guess consistent with the numerical data given by your calculator?