# Taking Sides

- Author:
- Judah L Schwartz

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 and its perimeter is . How do you convince a student that
the area of a circle is

*UNIT*circle is**half its perimeter?***NOT***What other questions [could,would] you ask you students based on this applet ?**Thanks to students in E330 for several good suggestions.