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Understanding Angles and Arcs in Circles

Learn about central angles, arcs, diameter, and circumference in circles. Discover how to calculate arc length and essential circle vocabulary.

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Understanding Angles and Arcs in Circles

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  1. Transparency 10-2 D B A C 5-Minute Check on Lesson 10-1 • Refer to ⊙F. • Name a radius • Name a chord • Name a diameter • Refer to the figure and find each measure • 4. BC • 5. DE • 6. Which segment in ⊙C is a diameter? FL, FM, FN, FO LN, MO, MN, LO LN, MO 3 13 Standardized Test Practice: A B C D D AB AC CD CB Click the mouse button or press the Space Bar to display the answers.

  2. Lesson 10-2 Angles and Arcs

  3. Objectives • Recognize major arcs, minor arcs, semicircles, and central angles and their measures • central angles sum to 360° • major arcs measure > 180° • minor arcs measure < 180° • semi-circles measure = 180° • Find arc length • Formula: C • (central angle / 360°) % of circle that is the arc

  4. Vocabulary • Central Angle – has the center of the circle as its vertex and two radii as sides • Arc – edge of the circle defined by a central angle • Minor Arc – an arc with the central angle less than 180° in measurement • Major Arc – an arc with the central angle greater than 180° in measurement • Semicircle – an arc with the central angle equal to 180° in measurement • Arc Length – part of the circumference of the circle corresponding to the arc

  5. y x Circles - Arcs Semi-CircleEHF Major Arc BEG E Diameter (d) Center CentralAngle B F BHG G Minor Arc H

  6. y x Circles - Probability Pie Charts Probability0 = no chance1 = for sure 90° 135° 135º------ = 3/8 360º or .375 or 37.5% Diameter (d) Radius (r) 0° 180° Center 45º------ = 1/8 360º or .125 or 12.5% 180º------ = 1/2 360º or .5 or 50% 315° 270° Circumference = 2πr = dπ

  7. Find . Example 2-1a ALGEBRA: Given Diameter RT

  8. The sum of the measures of Use the value of x to find Example 2-1b Substitution Simplify. Add 2 to each side. Divide each side by 26. Given Substitution Answer: 52

  9. ALGEBRA Refer to . Find . Example 2-1c

  10. form a linear pair. Linear pairs are supplementary. Substitution Simplify. Subtract 140 from each side. Answer: 40

  11. ALGEBRA AD and BE are diameters a. Find m b. Find m Example 2-1e Answer: 65 Answer: 40

  12. In bisects and is a minor arc, so is a semicircle. Find . Example 2-2a Answer: 90

  13. In bisects and since bisects . Find . is a semicircle. Example 2-2c Answer: 67

  14. In bisects and Find . Example 2-2e Answer: 316

  15. In and are diameters, and bisects Find each measure. a. b. c. Example 2-2g Answer: 54 Answer: 72 Answer: 234

  16. degree measure of arc In and . a) Find the length of . b) Find the length of arc DC. arc length circumference degree measure of whole circle In and . Write a proportion to compare each part to its whole. Example 2-4a

  17. Now solve the proportion for . Multiply each side by 9 . Answer: The length of is units or about 3.14 units. Example 2-4b Simplify. C ∙ (% of the circle) = 9π ∙ (140/360) = 7π/2 Answer: The length of arc DC is 7π/2 units or about 11 units.

  18. Summary & Homework • Summary: • Sum of measures of central angles of a circle with no interior points in common is 360° • Measure of each arc is related to the measure of its central angle • Length of an arc is proportional to the length of the circumference • Homework: • pg 533-534; 14-19; 24-29; 32-35

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