Arc length suggests a new way of measuring angles in circles. Previously, you have measured all angles in degrees, but you can also measure angles in radians. 1 radian is the angle that creates an arc that has a length equal to the radius. Below, the arc has a length equal to the radius. The angle that is created is 1 radian.
Therefore, you have the following conversions:
Note that if a given angle has no units, it is assumed to be in radians.
A sector is a portion of a filled circle bounded by two radii and an arc (a “wedge” of a circle).
Let's do a problem involving sector similarity.
Show that two sectors with the same central angle are similar by using transformations to explain why the two shaded sectors below are similar.
Next, let's look at a problem involving arc length.
In a circle of radius 1, the measure of a central angle in radians will be equal to the length of the intercepted arc. This is because the number of radians equals the number of radii that make up the arc.
Because the sectors are similar, corresponding lengths are proportional:
Finally, lets look at a problem where we measure arc length
Now you can find the length of the arc:
Earlier, you were asked about radians in cir
1. What's the difference between finding the measure of an arc and the length of an arc?
2. What is the relationship between a radian and a radius?
3. How are radians and degrees related?
4. Explain why the two shaded sectors below are similar.
15. Explain how to find the length of an arc when given the central angle in radians. How does this compare to the process of finding the length of an arc when given the central angle in degrees?
To see the Review answers, open this PDF file and look for section 8.9.