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 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 ?