In this assignment we will use Geometer's Sketchpad to discover relationships between the radius, circumference, and area of a circle. You may know already these relationships. If so, view this assignment as showing you why these relationships are true.
Open the file Circle.gsp in Geometer's Sketchpad. You will find a circle with its circumference, radius, and area calculated. Calculate the following ratios:
and
Do this by using the calculate tool (under the measure menu), and clicking on the circumference, radius, and area lines, respectively. Do not input the actual numbers!
Resize the circle. What happens to these two ratios? By creating the ratios in the manner described, you get the correct ratios for any sized circle, not just the original circle. Can you identify the first ratio in terms of a number with which you are familiar? Use the first ratio to find a formula for the circumference in terms of the radius. Then, use this formula and the second ratio to find a formula for the area in terms of just the radius, and not in terms of both the radius and circumference.