There is a circular path around a pond with a fountain in the center. The path is 2 miles long. When Tyler enters the path, he is east of the fountain. He starts walking counterclockwise. 

1. A sign says Tyler has walked half a mile. Sketch an image that shows the path, the fountain, the arc Tyler has walked, and the central angle defined by that arc.
2. Tyler is now northwest of the fountain. 
   1. What is the total distance Tyler has walked?
   2. What is the central angle defined by the arc Tyler has walked?
3. There is a duck statue  miles around the path. What is the central angle defined by the arc Tyler has walked when he reaches the statue?

Suppose we have a circle that has a central angle. The radian measure of the angle is the ratio of the length of the arc defined by the angle to the circle’s radius. That is, .

1. The image shows a circle with radius 1 unit.

   

   1. Cut a piece of string that is the length of the radius of this circle.
   2. Use the string to mark an arc on the circle that is the same length as the radius.
   3. Draw the central angle defined by the arc.
   4. Use the definition of a radian to calculate the radian measure of the central angle you drew.
2. Draw a 180-degree central angle (a diameter) in the circle. Use your 1-unit-long piece of string to measure the approximate length of the arc defined by this angle.
3. Calculate the radian measure of the 180-degree angle. Give your answer both in terms of and as a decimal rounded to the nearest hundredth.
4. Calculate the radian measure of a 360-degree angle.