If students struggle to begin, suggest they sketch some circles with sectors using both radians and degrees to use as examples. Also, ask students to find the radian measure of a full circle or a 360-degree angle: radians.

The goal is to discuss how radian measure makes for a useful sector area formula. Here are some questions for discussion:

• “Would this formula work if the angle were measured in degrees? Why or why not?” (No, it wouldn’t work. The in the denominator was the measure of the full circle in radians. If we were using degrees, that denominator would be 360, not .)
• “Is there a different formula for sectors with angles measured in degrees?” (Yes, we could write .)
• “Is the formula the only way to find the area of a sector with an angle measured in radians?” (No. We could use the method of finding the total area of the circle and the fraction represented by the sector, then multiplying those two together.)
• “What are the advantages and disadvantages of using this formula to calculate sector area?” (The formula simplifies some of the calculations. The formula is not very complicated. The formula works only for radians. The degree version is more complicated. We have to remember the formula correctly in order to use it.)

Students may use the distances between the curved paths as the circle's radii. Ask these students to imagine the full circle and then to figure out how far point is from the center.

The goal is to recognize that, because radian measure is defined as the constant of proportionality between arc length and radius, radian measure simplifies arc length calculations. First, ask students why a 1.5 radian measure of the angle in the activity makes sense. (The angle appears to measure a little less than radians. Dividing by 2 gives us about 1.57 radians, which is indeed a bit more than the 1.5 radians in the activity.)

Ask students to recall the definition of radian measure. () Invite them to solve this equation for . () Then display this equation for all to see:

Ask students what this equation represents. (It is a formula for arc length if the central angle is measured in degrees.) Then ask, “What are the advantages and disadvantages of degree and radian angle measurements?” Sample responses:

• There are easy formulas for sector area and arc length if we’re working with radians.
• Degrees may still be more familiar than radians.
• Radians often have the number in them, and they are often fractions, which may make them more difficult to work with.
• If we know an arc length and radius, we can directly calculate the radian measure of an angle. It’s more complicated to calculate the degree measure of the angle in this situation.