Chapter 6-1 Radians and Angles

Measure the length of a circle with this Interactive Diagram. You will see a tape measure already set up at point B on the circle. The circle's origin is point O. While "Resize circle" is NOT checked, Drag the handle at point A around the circle.

The tape measure is marked in radius lengths for the current circle.

The displays will keep track of the measure of the angle AOB, as well as the number of radians in the central angle corresponding to the arc from B to A, counterclockwise around the circle.

Change the size of the circle by checking in the "Resize circle" box and dragging on point A.

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Questions
1.How is the value of an angle measured in radians related to arclength?

2. Make a circle with a 2-cm. radius (relative to the scale bar). What is the central angle, in radians, for a semi-circle? Suppose you double the radius of the circle. What is the central angle, the new semi-circle, in radians? Explain.

3. Find the degree measure of an angle of 1 radian. Change the radius of the circle: what effect does the radius change have on the degree measure of the angle?

4. (a)What angle corresponds to an arc of /3 radians?
an arc of 8/3 radians? an arc of radians?

(b) Find the degree measure of an angle of 3.2 radians. Find the degree measure of an angle of 1.2 radians. What is the difference in the degree measurements?

(c) From these results, make a generalization about the relationship between angles measured in radians and angles measured in degrees.

5. What is the arclength in centimeters from point B to point A on the circle, if mAOB is 90°, and the radius is 2 cm.? If the radius of the circle were 5 cm., how long is the arc from point B to point A for the same angle?

6. Develop a general method for predicting the length of an arc from its central angle.