Circles

Lesson 11

Math

Unit 7

10th Grade

Lesson 11 of 14

Objective


Define, describe, and calculate arc length.

Common Core Standards


Core Standards

  • G.C.B.5 — Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Foundational Standards

  • 7.G.B.4
  • 7.G.B.5

Criteria for Success


  1. Define and describe arc length as a portion of the circumference.
  2. Derive a general formula to calculate the arc length based on the radius of a circle. 
  3. Distinguish between arc length (distance on the arc) and arc measure (angle measure of the arc that is congruent to the central angle that subtends the arc).
  4. Describe how the whole arc length and the arc measure of a circle are related when the radius of the circle is one. 

Tips for Teachers


  • This lesson will prepare students for the next lesson, which introduces radians. Therefore, it is important to ask students the relationship between arc length and arc measures in this lesson, which is Criteria for Success #4 and specifically addressed through a Guiding Question on Anchor Problem #2. 
  • In this lesson, make sure to ask students to both calculate out the value of pi and find the exact solution using pi in the answer.
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Anchor Problems


Problem 1

Which is the longer route?

Guiding Questions

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References

101Questions Rotonda West, FL

Rotonda West, FL by Dan Meyer is made available on 101Questions under the CC BY 3.0 license. Accessed Sept. 19, 2018, 2:16 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

What is the length of $$\widehat{BC}$$ in circle $$A$$?

Guiding Questions

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Target Task


Below is circle $$A$$, with radius $$\overline{AB}$$ measuring $$3$$ centimeters.

  1. What is the arc length of $$\widehat{BC}$$?
  2. What is the measure of $$\widehat{BC}$$?
  3. Explain how the arc length and arc measure are different.
  4. How are the arc length and circumference related? 

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students are asked to find the circumference based on the arc measure and arc length.
  • Include problems where students are asked to find the arc length and arc measure of concentric circles.
  • Include problems where students are asked to find the perimeter of composite figures that have partial circles as part of the diagram. 
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Lesson 10

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Lesson 12

Lesson Map

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Topic A: Equations of Circles

Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Topic C: Arc Length, Radians, and Sector Area

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