- Judah L Schwartz
Here is a regular polygon of n sides inscribed in a unit circle. In the limit of a very large number of sides the area and perimeter of the polygon approach those of the circle. Write an expression for A(n), the area of an n sided regular polygon inscribed in a unit circle. Write an expression for P(n), the perimeter of an n sided regular polygon inscribed in a unit circle. Contrast the rates at which A(n) and P(n) approach their limits. Challenges: The number of sides, n, grows while the length of each side, S gets smaller and smaller. How does the product of n and S behave? How do you know? Can you prove it? The area of a UNIT circle is and its perimeter is . How do you convince someone that its area is NOT half its perimeter? What other questions [could,would] you ask?