Circles

Lesson 14

Math

Unit 7

10th Grade

Lesson 14 of 14

Objective


Use sector area of circles to calculate the composite area of figures.

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.
  • N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
  • N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Foundational Standards

  • 7.G.B.4

Criteria for Success


  1. Identify the necessary formulas and measurements required to find the composite area using the sector area of circles. 
  2. Justify components of the formula and describe values of the components in the context of the problem. 
  3. Justify the level of precision used for the context of the problem and identify a reasonable range of solutions.
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Anchor Problems


Problem 1

Below is a diagram of a yard and its sprinkler system. The sprinklers are placed at point $$C$$ and point $$A$$ spraying in an arc as shown by the dark shaded area on the diagram. 

Assuming each unit is a foot, what is the area of the yard that is not watered by the sprinklers?

Guiding Questions

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References

GeoGebra Geometry - 8.14AP1

Geometry - 8.14AP1 by Match Fishtank is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 12:27 p.m..

Problem 2

Amy wanted to find the area that is the overlap of circles $$A$$ and $$C$$ below.

She started by drawing the sector $$EAC$$ from circle $$A$$ as shown below.

She saw that this did not cover the whole area that she wanted, so she drew sector $$ECA$$ from circle $$C$$ as shown below.

Amy said, "Aha! A triangle!" How does this help Amy find the area of the overlap of circles $$A$$ and $$C$$?

Guiding Questions

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References

GeoGebra Geometry - 8.14 AP2

Geometry - 8.14 AP2 by is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 12:28 p.m..

Target Task


Find the shaded area rounded to the nearest hundredth.

References

EngageNY Mathematics Geometry > Module 5 > Topic B > Lesson 10Exit Ticket, Question #2

Geometry > Module 5 > Topic B > Lesson 10 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

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 from the Problem Set Guidance in Lesson 13.
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Lesson 13

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

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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