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11-5

11-5. Area or Circles and Sectors. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. 11.5 Areas of Circles and Sectors. Warm Up 1. Find w , y , and z . Give the answers in simplest radical form. 11.5 Areas of Circles and Sectors. 2. RT. Warm Up Continued Find each measure.

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11-5

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  1. 11-5 Area or Circles and Sectors Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. 11.5 Areas of Circles and Sectors Warm Up 1.Find w, y, and z. Give the answers in simplest radical form.

  3. 11.5 Areas of Circles and Sectors 2. RT Warm Up Continued Find each measure. 54 3. UQ

  4. 11.5 Areas of Circles and Sectors Objectives Find the area of a circle and a sectors of a circle.

  5. 11.5 Areas of Circles and Sectors Vocabulary sector of a circle segment of a circle

  6. 11.5 Areas of Circles and Sectors You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram. The base of the parallelogram is about half the circumference, or r, and the height is close to the radius r. So A r · r = r2. The more pieces you divide the circle into, the more accurate the estimate will be.

  7. 11.5 Areas of Circles and Sectors

  8. 11.5 Areas of Circles and Sectors

  9. 11.5 Areas of Circles and Sectors The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the circle by

  10. 11.5 Areas of Circles and Sectors

  11. 11.5 Areas of Circles and Sectors Helpful Hint Write the degree symbol after m in the formula to help you remember to use degree measure not arc length.

  12. 11.5 Areas of Circles and Sectors Example 1A: Finding the Area of a Sector Find the area of each sector. Give answers in terms of and rounded to the nearest hundredth. sector HGJ Use formula for area of sector. Substitute 12 for r and 131 for m. Simplify. = 52.4 m2  164.62 m2

  13. 11.5 Areas of Circles and Sectors Example 1B: Finding the Area of a Sector Find the area of each sector. Give answers in terms of and rounded to the nearest hundredth. sector ABC Use formula for area of sector. Substitute 5 for r and 25 for m. Simplify.  1.74 ft2  5.45 ft2

  14. 11.5 Areas of Circles and Sectors Check It Out! Example 1a Find the area of each sector. Give your answer in terms of  and rounded to the nearest hundredth. sector ACB Use formula for area of sector. Substitute 1 for r and 90 for m. Simplify. = 0.25 m2  0.79 m2

  15. 11.5 Areas of Circles and Sectors Check It Out! Example 1b Find the area of each sector. Give your answer in terms of  and rounded to the nearest hundredth. sector JKL Use formula for area of sector. Substitute 16 for r and 36 for m. Simplify. = 25.6 in2  80.42 in2

  16. 11.5 Areas of Circles and Sectors Example 2: Automobile Application A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by the blade as it rotates through an angle of 122°? Use formula for area of sector. r = 18 in.  345 in2 Simplify.

  17. 11.5 Areas of Circles and Sectors Check It Out! Example 2 To the nearest square foot, what is the area watered in Example 2 (p. 765) as the sprinkler rotates through a semicircle? Use formula for area of sector. d = 720 ft, r = 360 ft  203,575 ft2 Simplify.

  18. 11.5 Areas of Circles and Sectors A segment of a circle is a region bounded by an arc and its chord.

  19. 11.5 Areas of Circles and Sectors Example 3: Finding the Area of a Segment Find the area of segment LNM to the nearest hundredth. Step 1 Find the area of sector LNM. Use formula for area of sector. Substitute 9 for r and 120 for m. Simplify. = 27 cm2

  20. 11.5 Areas of Circles and Sectors Step 2 Find the area of ∆LNM. Draw altitude NO. Example 3 Continued Find the area of segment LNM to the nearest hundredth. Simplify.

  21. 11.5 Areas of Circles and Sectors Remember! In a 30°-60°-90° triangle, the length of the leg opposite the 60° angle is √3 times the length of the shorter leg.

  22. 11.5 Areas of Circles and Sectors Example 3 Continued Find the area of segment LNM to the nearest hundredth. Step 3 area of segment = area of sector LNM – area of ∆LNM  49.75 cm2

  23. 11.5 Areas of Circles and Sectors Check It Out! Example 3 Find the area of segment RST to the nearest hundredth. Step 1 Find the area of sector RST. Use formula for area of sector. Substitute 4 for r and 90 for m. Simplify. = 4 m2

  24. 11.5 Areas of Circles and Sectors Check It Out! Example 3 Continued Find the area of segment RST to the nearest hundredth. Step 2 Find the area of ∆RST. ST = 4 m, and RS = 4m. = 8 m2 Simplify.

  25. 11.5 Areas of Circles and Sectors Check It Out! Example 3 Continued Find the area of segment RST to the nearest hundredth. Step 3 area of segment = area of sector RST – area of ∆RST = 4 – 8  4.57 m2

  26. 11.5 Areas of Circles and Sectors In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle.

  27. 11.5 Areas of Circles and Sectors Example 4A: Finding Arc Length Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. FG Use formula for area of sector. Substitute 8 for r and 134 for m.  5.96 cm  18.71 cm Simplify.

  28. 11.5 Areas of Circles and Sectors Example 4B: Finding Arc Length Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. an arc with measure 62 in a circle with radius 2 m Use formula for area of sector. Substitute 2 for r and 62 for m.  0.69 m  2.16 m Simplify.

  29. 11.5 Areas of Circles and Sectors =  m  4.19 m Check It Out! Example 4a Find each arc length. Give your answer in terms of and rounded to the nearest hundredth. GH Use formula for area of sector. Substitute 6 for r and 40 for m. Simplify.

  30. 11.5 Areas of Circles and Sectors Check It Out! Example 4b Find each arc length. Give your answer in terms of and rounded to the nearest hundredth. an arc with measure 135° in a circle with radius 4 cm Use formula for area of sector. Substitute 4 for r and 135 for m. = 3 cm  9.42 cm Simplify.

  31. 11.5 Areas of Circles and Sectors Lesson Quiz: Part I Find each measure. Give answers in terms of and rounded to the nearest hundredth. 1. area of sector LQM 7.5in223.56 in2 2. length of NP 2.5 in. 7.85 in.

  32. 11.5 Areas of Circles and Sectors Lesson Quiz: Part II 3. The gear of a grandfather clock has a radius of 3 in. To the nearest tenth of an inch, what distance does the gear cover when it rotates through an angle of 88°? 4.6 in. 4. Find the area of segment GHJ to the nearest hundredth. 55.94 m2

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