Lesson 13

In this activity, students find the fraction of a circle represented by a central angle, and they use this fraction to find the area of the sector. The primary goal of the activity is to elicit students’ strategies for finding the fraction. Monitor for strategies such as converting the radian measure to degrees, drawing diagrams, and dividing \(\frac{\pi}{3}\) by \(2\pi\).

Invite previously selected students to explain how they found the fraction \(\frac16\). Sample responses:

• I converted the radian measure \(\frac{\pi}{3}\) to 60 degrees. I know that 60 degrees is \(\frac16\) of 360.
• I drew a diagram of a circle. I know half the circle is \(\pi\) radians, and \(\frac{\pi}{3}\) is \(\frac13\) of that.
• I divided \(\frac{\pi}{3}\) by \(2\pi\). That is the same as multiplying \(\frac{\pi}{3}\) by \(\frac{1}{2\pi}\).

Students combine their previous work with sector area and their new knowledge of radians to justify a formula for the area of a sector.

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: \(2\pi\) 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 \(2\pi\) in the denominator was the measure of the full circle in radians. If we were using degrees, that denominator would be 360, not \(2\pi\).)
• “Is there a different formula for sectors with angles measured in degrees?” (Yes, we could write \(\frac{\theta}{360}\boldcdot \pi r^2\).)
• “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 multiply those two together.)
• “What are the advantages and disadvantages of using this formula to calculate sector area?” (Sample responses: The formula simplifies some of the calculations. The formula is not very complicated. The formula only works for radians. The degree version is more complicated. You have to remember the formula correctly in order to use it.)

Students revisit the idea of radians as the constant of proportionality between arc length and radius for a given central angle. They observe that radian measure allows them to make simple calculations for arc length.

Students may not be certain that radius is the independent or \(x\)-variable.

Students may use the distances between the exits as the circle radii. Ask these students how far point \(B\) is from town hall.

Some students may not recognize 1.5 as a radian measure because it doesn’t have the form of a fraction involving \(\pi\). Remind these students that radians measures like \(2\pi\) and \(\frac{\pi}{4}\) are indeed numbers on the number line. Invite them to calculate the decimal measures of several radian values to convince themselves that 1.5 radians is a reasonable value for the angle measure in the image.

The goal is to recognize that because radian measure is defined as the constant of proportionality between arc length and radius, it 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 \(\frac{\pi}{2}\) radians. Dividing \(\pi\) by 2 gives us about 1.57 radians, which is indeed a bit more than the 1.5 radians in the activity.)

First, ask students to recall the definition of radian measure (\(\theta = \frac{\ell}{r}\)). Invite them to solve this equation for \(\ell\) (\(\ell=\theta\boldcdot r\)). Next, display this equation for all to see:

\(\ell = \frac{d}{360}\boldcdot 2\pi r\)

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 \(\pi\) 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.