The figure shows part of a regular n-gon inscribed in a circle of radius r. Use the figure to answer the questions and derive a formula for the circumference of a circle with radius r.
1. The regular n-gon can be divided into n congruent isosceles triangles, each with a vertex angle of measure. What is the measure of 360 degrees/n in terms of n? Explain.
2. Use the sine ratio to write an expression for x in terms of r and n. Show your work.
3. Use the expression you found for x to write an expression for the perimeter of the inscribed n-gon.
4. Your expression for the perimeter of the inscribed n-gon should contain the factor n sin (180 degrees/n). Use technology to help you explain what happens to the value of n sin (180 degrees/n) as n gets larger and larger. (Hint: Describe how this information relates to the circumference of a circle.)
5. As n gets larger and larger, the perimeter of the inscribed n-gon gets closer and closer to the circle. What does the expression in question 2 and your answer to question 4 tell you about the formula for the circumference of a circle? Explain.