# Lesson 13

## 13.1: What Fraction? (5 minutes)

### Warm-up

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 $$2\pi$$.

### Student Facing

A circle with radius 24 inches has a sector with central angle $$\frac{\pi}{3}$$ radians.

1. What fraction of the whole circle is represented by this sector?
2. Find the area of the sector.

### Activity Synthesis

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 $$\frac{1}{2\pi}$$.

## 13.2: A Sector Area Shortcut (15 minutes)

### Activity

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

### Launch

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Prior to the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their explanation of why the formula Elena saw works. Display prompts for feedback that students can use to help their partner strengthen and clarify their ideas. For example, "Your explanation tells me . . .", "Can you say more about why you . . . ?", and "A detail (or word) you could add is _____, because . . . ." Give students 2–3 minutes to revise their initial draft based on feedback from their peers. This will help students use precise language to justify the formula for the area of a sector.
Design Principle(s): Optimize output (for justification); Cultivate conversation

### Student Facing

Lin and Elena are trying to find the area of the shaded sector in the image. Lin says, “We’ve found sector areas when the central angle is given in degrees, but here it’s in radians. Should I start by finding the area of the full circle?”

Elena says, “I saw someone using the formula $$\frac12 r^2 \theta$$ where $$\theta$$ is the measure of the angle in radians, and $$r$$ is the radius. But I don’t know where that came from.”

1. Compare and contrast finding sector areas for central angles measured in degrees and those measured in radians.
2. Explain why the formula that Elena saw works.
3. Find the area of the sector.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.)

## 13.3: An Arc Length Shortcut (15 minutes)

### Activity

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.

### Launch

Reading, Listening, Conversing: MLR6 Three Reads. Use a modified version of this routine to support reading comprehension of this problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about. (The city of Riverside has two straight highways and several roads shaped like circular arcs connected to the highways.) Use the second read to identify important features and relationships. Listen for and amplify important quantities that vary in relation to each other in this situation: radius, $$r$$, of the highway and arc length, $$l$$, of the connector roads. After the third read, ask students to brainstorm how to create a table for the first question. This will help students connect the language in the word problem and the reasoning needed to create a table while keeping the cognitive demand of the task.
Design Principle(s): Support sense-making
Action and Expression: Internalize Executive Functions. Provide students with a blank table with two columns: $$r$$ (miles) and $$\ell$$ (miles). Ask what elements are required to write the equation of a line (the slope and the $$y$$-intercept or any other point on the line).
Supports accessibility for: Language; Organization

### Student Facing

The city of Riverside has 2 straight highways extending out from its town hall (located at $$T$$ in the image). Several roads shaped like circular arcs connect the two highways. Exits from 1 of the highways are shown at points $$A,B,C,$$ and $$D$$.

1. Create a table that shows the arc length, $$\ell$$, of the connector roads as a function of the radius, $$r$$, of the highway.
2. Plot the points from your table on the coordinate grid and connect them.
3. The points should form a line. Write an equation for this line, using the variables $$\ell$$ and $$r$$.
4. What does the slope of the line mean in context of the highways and connector roads?
5. The city wants to build a new arc-shaped connector road starting at point $$E$$ which will be an additional 6 miles past exit $$D$$. How long will this road be?

### Student Facing

#### Are you ready for more?

Suppose a Ferris wheel with a 50 foot radius takes 2 minutes to complete 1 rotation.

1. After 20 seconds, through how many radians would a rider on the Ferris wheel have traveled?
2. How long would it take to travel through 3 radians?
3. What is the rotational speed of the rider on the ferris wheel in radians per second?
4. A decorative light is attached to the Ferris wheel 25 feet from the center of the wheel. What is the rotational speed of the light in radians per second?

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

## Lesson Synthesis

### Lesson Synthesis

The goal is to summarize what students know about radian angle measures. Arrange students in groups of 4 and give them 3 minutes to record as many facts about radians as they can. Sample responses:

• The definition of radian measure is the ratio of arc length to radius.
• A complete circle contains 360 degrees, which is $$2\pi$$ radians.
• A formula for sector area is $$\frac12 r^2 \theta$$.
• A formula for arc length is $$\ell=r\theta$$.
• The degree measure 180 degrees is equivalent to $$\pi$$ radians.
• Radians measure how many radii fit around the arc length for a given central angle.
• Radians are the constant of proportionality between arc length and radius.

## 13.4: Cool-down - Calculate It (5 minutes)

### Cool-Down

Suppose we want to find the area of a sector of a circle whose central angle is $$\theta$$ radians and whose radius is $$r$$ units. First we find the area of the whole circle, or $$\pi r^2$$ square units. Then we find the fraction of the circle represented by the sector. The complete circle measures $$2\pi$$ radians. So the fraction is $$\frac{\theta}{2\pi}$$. Multiply the fraction by the area to get $$\frac{\theta}{2\pi}\boldcdot \pi r^2$$. This can be rewritten as $$\frac12 r^2 \theta$$.
For example, let the radius be 25 units and the radian measure of the sector’s central angle be $$\frac{3\pi}{5}$$ radians. Substitute these values into the formula we just created to get $$\frac12 (25)^2 \boldcdot \frac{3\pi}{5}$$. This can be rewritten as $$\frac{375\pi}{2}$$, which is about 589 square units. Using the formula was a bit quicker than going through the process of finding the specific fraction of the circle represented by this sector and multiplying by the circle’s area.
Additionally, we can write a formula for arc length based on the definition of radian measure as $$\theta=\frac{\ell}{r}$$ where $$\theta$$ is the central angle measure in radians, $$\ell$$ is the arc length, and $$r$$ is the radius. Rewrite the definition to solve for arc length. We get $$\ell=r\theta$$.