# Lesson 10

Angles, Arcs, and Radii

## 10.1: Comparing Progress (5 minutes)

### Warm-up

In this activity, students continue to explore the relationship between arc length, radius, and central angle. They recognize that a particular central angle defines arc lengths of different sizes in circles with different radii. This idea leads in to a subsequent activity in which students compare ratios of arc length to radius for angles and circles of different sizes.

### Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

### Student Facing

Han and Tyler are each completing the same set of tasks on an online homework site. Han is using his smartphone and Tyler is using his tablet computer. Their progress is indicated by the circular bars shown in the image. The shaded arc represents the tasks that have been completed. When the full circumference of the circle is shaded, they will be finished with all the tasks.

Tyler says, “The arc length on my progress bar measures 4.75 centimeters. The arc length on Han’s progress bar measures 2.25 centimeters. So, I’ve completed more tasks than Han has.”

- Do you agree with Tyler? Why or why not?
- What information would you need to make a completely accurate comparison between the two students’ progress?

### Student Response

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### Activity Synthesis

The goal of the discussion is to start to build the idea that the length of an arc and the radius of the circle give us information about the central angle that defines the arc. Ask students these questions:

- “There are actually 2 circles that make up each progress bar, an inner circle and an outer circle. Which one should we focus on when measuring the progress?” (It doesn’t matter; either contains the same information.)
- “If the length of the arc on Tyler’s circle is longer, why does that not necessarily mean he has completed more tasks?” (The completed proportion depends not only on the arc length but on the size of a circle.)
- “For the second question, what information do you think would be helpful to accurately compare the progress of the two students?” (Students may discuss proportions and fractions. If the phrase
*central angle*does not come up, ask students what information this measurement would give us.) - “What measurement can we use to compare the ‘sizes’ of the circle?” (Radius gives relative size.)

## 10.2: A Dilated Circle (10 minutes)

### Activity

In this activity, students revisit the fact that when a figure is dilated, corresponding angles do not change measure, all lengths and distances change by the scale factor, and areas change by the square of the scale factor. They are also reminded that all circles are similar, so any one circle is a dilation of any other circle. Students note that the ratio of a circle’s circumference to its diameter is a constant, and that this is the origin of the circumference formula.

These concepts will be useful in subsequent activities in which students determine that arc length to radius ratios are constant for given central angles.

### Launch

*Action and Expression: Internalize Executive Functions.*Provide students with a table to organize example calculations. Invite them to choose a value for the small radius and use it throughout, completing the questions in any order they like. Label the columns: part, small, large, and factor by which it’s changed. Label the rows: sector area, central angle, radius, arc length, and ratio of circumference to diameter.

*Supports accessibility for: Language; Organization*

### Student Facing

The image shows 2 circles. The second circle is a dilation of the first circle using a scale factor of 3.

For each part of the dilated image, determine the factor by which it’s changed when compared to the corresponding part of the original circle.

- the area of the sector
- the central angle of the sector
- the radius
- the length of the arc defined by the sector
- the ratio of the circle’s circumference to its diameter

### Student Response

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### Anticipated Misconceptions

If students aren’t sure how to find the ratio of the circles’ circumferences and diameters, remind them that circumference can be calculated with the formula \(C=2\pi r\). Ask if this formula can be written in terms of diameter instead of radius.

### Activity Synthesis

The goal is to discuss ratios that arise from the similarity of all circles. Ask students:

- “Suppose we draw a tiny circle with a 90 degree sector. Is it a dilation of the original circle? Why or why not?” (Yes. All circles are similar, so they are all dilations of each other.)
- “Suppose we dilated the circle not by a factor of 3, but by a factor of 1,000,000. What would happen to the radius, arc length, and central angle, and why?” (The radius and arc length would increase by a factor of 1,000,000, because distances are multiplied by the scale factor under dilation. Angles don’t change under dilation, so the central angle would still measure 90 degrees.)
- “What would be the ratio of our enormous circle’s circumference to its diameter?” (It would still be \(\pi\).)
- “How does this constant ratio help us when we’re calculating circle circumferences?” (Because the circumference divided by the diameter always equals \(\pi\), we can multiply the diameter by \(\pi\) to find a circle’s circumference.)
- “In what other context did we talk about ratios that didn’t change, no matter the size of the object?” (The trigonometric ratios were like this.)

## 10.3: Card Sort: Angles, Arcs, and Radii (20 minutes)

### Activity

A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). In this task, students examine relationships between arc lengths, radii, and central angles. They observe that the ratio between arc length and radius appears to be constant for a given central angle, and may be a proxy for angle measurement.

In a subsequent activity, students will prove that the ratio of arc length to radius is indeed constant for a particular central angle regardless of the size of the circle, and they’ll define this ratio as the radian measure of the angle.

Monitor for groups who recognize that central angles can be calculated for the measurements given on cards B, C, G, and H, and for those who consider ratios between the arc lengths and the radii. It’s appropriate for students’ thinking to be informal at this point (for example, they may notice that the integer portion of the arc length is \(\frac14\) the radius for cards B and H).

### Launch

Arrange students in groups of 2. Distribute pre-cut slips. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the cards, they should work with their partner to come up with categories.

Give students 7–8 minutes of work time, then follow with a whole-class discussion. Be sure to leave enough time for students to calculate arc length to radius ratios in the synthesis.

*Conversing: MLR2 Collect and Display.*As students sort the cards into categories, listen for and collect the language students use to categorize cards B, C, G, and H. Capture student language that suggests that the arc length and radii measurements are related to the central angle of 45 or 270 degrees. Write the students’ words and phrases on a visual display. As students review the visual display, ask students to revise and improve how ideas are communicated. For example, a statement such as, “The numbers divide to equal \(\frac{\pi}{4}\)” can be improved with the statement, “The central angle of 45 degrees has an arc length to radius ratio of \(\frac{\pi}{4}\).” This will help students use the mathematical language necessary to precisely describe the relationship between central angles, arc lengths, and radii.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

*Representation: Internalize Comprehension.*Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed sorting their initial set.

*Supports accessibility for: Conceptual processing; Organization*

### Student Facing

Your teacher will give you a set of cards. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into categories in a different way. Be prepared to explain the meaning of your new categories.

### Student Response

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### Student Facing

#### Are you ready for more?

For a circle of radius \(r\), an expression that relates the area of a sector to the arc length defined by that sector is \(A = \frac12 r \ell\) where \(A\) is the area of the sector and \(\ell\) is the length of the arc. Explain why this is true and provide an example.

### Student Response

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### Anticipated Misconceptions

If no groups appear to be creating 2 categories of congruent angles, ask students what they can figure out about the circles on the cards that show arc lengths and radius measurements.

### Activity Synthesis

Select previously identified groups to share their categories and how they sorted their cards. If possible, choose a group that created a group of 45 degree angles (cards B, D, F, and H) and another group of 270 degree angles (cards A, C, E, and G). Attend to the language that students use to describe their categories, giving them opportunities to describe the categories more precisely. Highlight the use of terms like *central angle* and *ratio*.

Then, if students have not already done so, ask them to calculate the central angle and the ratio of arc length to radius for each of cards B, C, G, and H. If time is short, consider dividing the problems amongst the class so each student has only 1 problem to work. Invite them to share observations about their results. Be sure the following points come up in the discussion:

- Cards B and H each represent 45 degree angles and have arc length to radius ratios equivalent to \(\frac{\pi}{4}\) or \(0.25\pi\).
- Cards C and G each represent 270 degree angles and have arc length to radius ratios equivalent to \(\frac{3\pi}{2}\) or \(1.5\pi\).
- The ratio appears to be the same for a particular central angle.

Ask students, “Suppose I gave you another card that showed an arc length of \(30\pi\) units and a radius of 20 units. Without actually doing a calculation, what central angle measure do you think might go with that, and why?” (The ratio of arc length to radius is equivalent to \(\frac{3\pi}{2}\), so it is probably a 270 degree angle.)

## Lesson Synthesis

### Lesson Synthesis

The goal is to discuss the fact that the ratio of the arc length and the radius of a central angle in a circle gives us information about the measurement of the central angle. Ask students:

- “Suppose a circle has a radius of 8 units and an arc with length \(2\pi\) units. What is the ratio of the arc length to the radius?” (The ratio is \(\frac{2\pi}{8}\), which is equivalent to \(\frac{\pi}{4}\).)
- “Based on the work you did in the card sort, what might be the measure of the central angle?” (It is probably 45 degrees. The 2 ratios that came out to \(\frac{\pi}{4}\) were from arcs created by a central angle measuring 45 degrees.)
- “Use another method to verify the measure of the angle.” (The circumference of the circle is \(16\pi\) units. The arc length of \(2\pi\) units is \(\frac18\) of that. So the angle measure must be 45 degrees because \(360 \div 8=45\).)

Tell students that this ratio, then, helps us understand the size of the corresponding angle. Display this image for all to see, explaining that each represents data about a central angle in a circle. Ask students to put the angles in order of size from smallest to largest, without actually calculating the degree measure of the angle.

angle name | circle radius | arc length |
---|---|---|

\(A\) | 10 units | \(5\pi\) units |

\(B\) | 12 units | \(2\pi\) units |

\(C\) | 20 units | \(30\pi\) units |

Angle \(B\) is the smallest, with an arc length to radius ratio of \(\frac{\pi}{6}\). Angle \(A\) is next, with a ratio of \(\frac{\pi}{2}\). Angle \(C\) is the largest, with a ratio of \(\frac{3\pi}{2}\).

## 10.4: Cool-down - Comparing Angles (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

If we have the same central angle in 2 different circles, the length of the arc defined by the angle depends on the size of the circle. So, we can use the relationship between the arc length and the circle’s radius to get some information about the size of the central angle.

For example, suppose circle A has radius 9 units and a central angle that defines an arc with length \(3\pi\). Circle B has radius 15 units and a central angle that defines an arc with length \(5\pi\). How do the 2 angles compare?

For the angle in Circle A, the ratio of the arc length to the radius is \(\frac{3\pi}{9}\), which can be rewritten as \(\frac{\pi}{3}\). For the angle in Circle B, the arc length to radius ratio is \(\frac{5\pi}{15}\), which can also be written as \(\frac{\pi}{3}\). That seems to indicate that the angles are the same size. Let’s check.

Circle A’s circumference is \(18\pi\) units. The arc length \(3\pi\) is \(\frac16\) of \(18\pi\), so the angle measurement must be \(\frac16\) of 360 degrees, or 60 degrees. Circle B’s circumference is \(30\pi\) units. The arc length \(5\pi\) is \(\frac16\) of \(30\pi\), so this angle also measures \(\frac16\) of 360 degrees or 60 degrees. The 2 angles are indeed congruent.