I saw the following question shared on twitter by @ddmeyer. I decided to take a closer look with GeoGebra. Scroll down the page until you see the graph below then drag the large red point on the pop-up.
Be sure to disable the pop-up blocker, at least temporarily, if you don't see it. The pop-up is needed for the applet.
Increase the number of sides n by dragging the slider then drag the large red point around again. Also try out the circle. If you accidentally close the pop-up, click on "View" then on "Graphics 2" and the pop-up should return. If you'd like to save a copy of applet simply click on "File" then "Save" or "Save As" and select a filename and location.
Created with GeoGebra
Extension (Pre-Calc): Find the general equation for area A in terms of perimeter P given the number of sides n. Is this function linear, quadratic, cubic, exponential, or does it use some other model?
Extension (Calculus): What do you notice as n increases? Conjecture about what happens to the curve as n increases without bound?