Regular Polygon and Circle Area

Area, Circle
How can we use the regular polygon area formula "A=1/2(apothem)(perimeter)" to find the formula for the area of a circle?
1. What happens to the appearance of the regular polygon as the number of sides increases? 2. What happens to the length of the apothem as the number of sides increases? What length does it approach? 3. What type of relationship exists between the number of sides and the length of each side? 4. What does the perimeter of the polygon appear to approach as the number of sides becomes larger? 5. Write the equation for the area of the polygon if the number of sides is infinite: