# Taking Sides

A circle of radius 1 circumscribes a regular polygon of n sides. Inside the regular polygon is an inscribed circle. In the limit of a very large number of sides the area and perimeter of both the inner and outer circles approach those of the polygon. Write an expression for , the area of an n sided regular polygon inscribed in a unit circle. Write an expression for , the perimeter of an n sided regular polygon inscribed in a unit circle. Write an expression for , the area of an n sided regular polygon that circumscribes a unit circle. Write an expression for , the perimeter of an n sided regular polygon that circumscribes a unit circle. Contrast the rates at which the areas and the perimeters 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 a student that the area of a circle is NOT half its perimeter? What other questions [could,would] you ask you students based on this applet ?
Thanks to students in E330 for several good suggestions.