Regular Polygon in a Circle

Topic:
Circle
Enter a number of sides (from 3 to 360), use the slider, or use the next and prev buttons to inscribe a regular polygon in the circle of radius 7 provided. The Apothem (the dashed line in the applet below) is the length from the center of the regular to one of its sides. Interact with this applet to get familiar with the concepts. One particular concept to take note is the relationship between the length of the perimeter of a regular polygon, and its apothem. (Or, rather, twice its apothem) When you are ready, answer the questions below.

Start with 3 on the slider. What shape does it make?

The big triangle is split into how many smaller congruent triangles?

Those smaller triangles are isosceles. Please explain why.

What part of the triangle is the apothem?

Click next. It should be a square now. How many smaller isosceles triangle do you see now? How do you know that they are isosceles?

Now, keep clicking "next". Notice the relationship between the apothem and the radius of the circle. Describe, in your own words, the relationship between the apothem and the radius of the circle as the sides of the regular polygon increase.