Lesson 8

Arcs and Sectors

8.1: Math Talk: Fractions of a Circle (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for calculating areas of sectors and finding arc lengths for simple fractions of circles. These understandings help students develop fluency and will be helpful later in this lesson when students will develop general methods for calculating sector areas and arc lengths.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they combine fraction operations with area and circumference formulas.

Launch

If necessary, provide students with formulas for the area and circumference of a circle.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Evaluate each problem mentally.

• Find the area of the shaded portion of the circle.
• Find the length of the highlighted portion of the circle’s circumference.
• Find the area of the shaded portion of the circle.
• Find the length of the highlighted portion of the circle’s circumference.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

When calculating the areas, students may divide the radius in half before applying the area formula, instead of first calculating the area of the entire circle then dividing it in half. Ask these students to calculate $$\pi \boldcdot \left(8^2\right)$$ then divide by 2, and compare this to the result of $$\pi \boldcdot 4^2$$.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .”  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

8.2: Sector Areas and Arc Lengths (15 minutes)

Activity

In this activity, students get experience calculating areas of circle sectors and lengths of arcs. This will help them define general methods for these calculations in a subsequent activity. In the activity synthesis, students observe that sectors can be used to create an informal argument for the area of a circle.

Monitor for a variety of strategies on the last problem, in particular for students who recognize that a 135 degree sector is $$\frac38$$ of a circle and for those who divide 135 by 360 to get the decimal 0.375.

Launch

After students have read the definition of a sector, tell them that any 2 radii define 2 mutually exclusive sectors. Shading or description will denote the particular sector that students should work with.

Action and Expression: Internalize Executive Functions. Provide students with a table with four columns labeled diagram, angle, area, and arc. Consider including columns for calculations or fraction of the circle for students who need extra support.
Supports accessibility for: Language; Organization

Student Facing

A sector of a circle is the region enclosed by 2 radii.

For each circle, find the area of the shaded sector and the length of the arc that outlines the sector. All units are centimeters. Give your answers in terms of $$\pi$$.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

1. What length of rope would you need to wrap around Earth’s equator exactly once?
2. Suppose you lengthen the rope, then suspend it so it is an equal distance away from Earth at all points around the equator. How much rope would you have to add in order to allow a person to walk underneath it?
3. What length of rope would you need to wrap around the circumference of a hula hoop with radius 70 centimeters exactly once?
4. Suppose you lengthen the rope, then lay the hula hoop and the rope on the floor, with the rope arranged in a circle with the same center as the hula hoop. How much rope would you have to add in order to allow a person to lie down on the floor with their toes pointing towards the hula hoop and their head towards the rope, but touching neither?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

If students are stuck on the last circle, ask them how they approached the first problems, and if any concepts carry over. Remind them that 360 degrees defines a complete circle, and ask if that number can help them.

Activity Synthesis

Select previously identified students to share their strategies for the final problem in this order: first, a student who recognized that a 135 degree sector is $$\frac38$$ of a circle; second, a student who divided 135 by 360 to get 0.375. Ask students the advantages and disadvantages of each approach. The second approach works for any central angle, while the first approach may be faster. The first approach may lead to an intuitive estimate that allows for easy recognition of calculation errors, and it always results in an exact answer that does not require approximation or rounding.

Next, tell students that we can use sectors to help create the original circle area formula. Display this image for all to see.

Ask students, “How does this image explain the formula for the area of a circle?” (If we slice the circle into many sectors and rearrange them, they take on a shape resembling a parallelogram. The parallelogram has height $$r$$. The length of the parallelogram is half the circumference of the circle, or $$\pi r$$. Multiply the base and height of the parallelogram to get $$\pi r^2$$ for its area.)

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. As students explain how the image supports the formula for the area of the circle, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement such as “sector,” “radius,” “circumference,” “parallelogram,” “base,” and “height.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

8.3: Build a Method (15 minutes)

Activity

In this activity, students generalize their earlier experiences with calculating sector areas and arc lengths.

Monitor for students who describe their method using words and for those who create a formula.

Student Facing

Mai says, “I know how to find the area of a sector or the length of an arc for central angles like 180 degrees or 90 degrees. But I don’t know how to do it for central angles that make up more complicated fractions of the circle.”

1. In the diagram, the sector’s central angle measures $$\theta$$ degrees and the circle’s radius is $$r$$ units. Use the diagram to tell Mai how to find the area of a sector and the length of an arc for any angle and radius measure.

2. This image shows a circle with radius and central angle measurements. Find the area of the shaded sector, and the length of the arc defined by the sector.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Choose previously identified students to share their strategies in this order: first, a student who gave a description in words; next, one who created a formula such as $$\frac{\theta}{360}\boldcdot \pi r^2$$. If no student created a formula, invite students to rewrite their method using symbols instead of words.

Challenge students to explain why the formula and the verbal description are expressing the same process. Ask, “What does the $$\frac{\theta}{360}$$ represent?” (It represents the fraction of the circle taken up by the sector.) “How about the $$\pi r^2$$?” (That’s the area of the whole circle.)

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares their strategy for finding the area of a sector or the length of an arc, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “Find the area of the circle,” can be restated as a question such as, “Do you agree that we need to find the area of the circle in order to find the area of the shaded sector?”
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

In this lesson, students solved problems involving arc length and sector area. Here are some questions for discussion:

• “Compare and contrast the processes for finding the area of a sector and calculating the length of an arc.” (Each calculation involves finding the fraction of a circle represented by a sector. For arc length, the fraction is multiplied by the circle’s circumference. For sector area, the fraction is multiplied by the circle’s area.)
• “What are some examples in the real world of arcs or sectors?” (Sample responses: A slice of pizza, the area wiped by a windshield wiper, and the wedges of a dartboard are shaped like sectors. Arc length can describe the path of an object in circular motion, like the orbit of a satellite around the earth or the path of someone riding a Ferris wheel. It’s also the distance around the circular portion of a racetrack.)

Finally, ask students to add this theorem to their reference charts as you add it to the class reference chart. Tell students that this wording is just a suggestion. They should add the method that they used in this lesson.

To calculate the area of a sector or the length of an arc, first find the fraction of the circle represented by the central angle of the arc or sector. Multiply this fraction by the circle’s area or circumference. (Theorem)

• arc length: $$3\pi$$ cm
• sector area: $$6\pi$$ cm2

8.4: Cool-down - Use Your Method (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

A sector of a circle is the region enclosed by 2 radii. To find the area of a sector, start by calculating the area of the whole circle. Divide the measure of the central angle of the sector by 360 to find the fraction of the circle represented by the sector. Then, multiply this fraction by the circle’s total area. We can use a similar process to find the length of the arc lying on the boundary of the sector.

The circle in the image has a total area of $$144\pi$$ square centimeters, and its circumference is $$24\pi$$ centimeters. To find the area of the sector with a 225$$^\circ$$ central angle, divide 225 by 360 to get $$\frac58$$ or 0.625. Multiply this by $$144\pi$$ to find that the area of the sector is $$90\pi$$ square centimeters. The length of the arc defined by the sector is $$15\pi$$ because $$24\pi \boldcdot \frac58 = 15\pi$$.