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Warm Up

Learn how to find the circumference and arc length of a circle using the formulas C = πd or C = 2πr, and the ratio of an arc's length to the circumference. Explore examples and solve problems involving finding the length of arcs. Also, discover how to find the area of circles and sectors using the formulas A = πr² and A = (m/360)πr².

hillary-knowles
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Warm Up

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  1. Warm Up

  2. 11.6 Circumference and Arc Length

  3. Finding circumference and arc length • The circumference of a circle is the distance around the circle. For all circles, the ratio of the circumference to the diameter is the same. This ratio is known as  or pi.

  4. The circumference C of a circle is C = d or C = 2r, where d is the diameter of the circle and r is the radius of the circle. Theorem 11.6: Circumference of a Circle

  5. And . . . • An arc length is a portion of the circumference of a circle. You can use the measure of an arc (in degrees) to find its length (in linear units).

  6. Arc Length Corollary • In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°. Arc length m = 2r 360° m Arc length = • 2r 360°

  7. The length of a semicircle is half the circumference, and the length of a 90° arc is one quarter of the circumference. More . . . ½ • 2r r r ¼ • 2r

  8. Ex. 2: Finding Arc Lengths • Find the length of each arc. a. b. c. 50° 100° 50°

  9. • 2(5) a. Arc length of = 50° 360° Ex. 2: Finding Arc Lengths • Find the length of each arc. # of ° a. • 2r a. Arc length of = 360° 50°  4.36 centimeters

  10. Ex. 2: Finding Arc Lengths • Find the length of each arc. # of ° b. • 2r b. Arc length of = 360° 50° 50° • 2(7) b. Arc length of = 360°  6.11 centimeters

  11. Ex. 2: Finding Arc Lengths • Find the length of each arc. # of ° c. • 2r c. Arc length of = 360° 100° 100° • 2(7) c. Arc length of = 360°  12.22 centimeters

  12. m Ex. 3: Using Arc Lengths • Find the indicated measure. a. circumference Arc length of = 2r 360° 3.82 = 60° 2r 360° 60° 3.82 1 = 2r 6 3.82(6) = 2r 22.92 = 2r C = 2r; so using substitution, C = 22.92 meters.

  13. 10-7 Area of Circles and Sectors

  14. Ex. 3: Using Arc Lengths • Find the indicated measure. b. m Arc length of m = 2r 360° 18 m = 360° • • 360° 2(7.64) 360° 135°  m

  15. Theorem 10-11: Area of a Circle • The area of a circle is  times the square of the radius or A = r2.

  16. Find the area of P. Solution: Use r = 8 in the area formula. A = r2 =  • 82 = 64  201.06 Ex. 1: Using the Area of a Circle • So, the area if 64, or about 201.06 square inches.

  17. Find the diameter of Z. Solution: Area of circle Z is 96 cm2. A = r2 96=  r2 96= r2  30.56  r2 5.53  r Ex. 1: Using the Area of a Circle • The diameter of the circle is about 11.06 cm.

  18. Theorem 10.12: Area of a Sector • The ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360°. m m A = or A = • r2 r2 360° 360°

  19. Ex. 2: Finding the area of a sector • Find the area of the sector shown below. Sector CPD intercepts an arc whose measure is 80°. The radius is 4 ft. m A = • r2 360°

  20. m A = • r2 360° 80° = • r2 360° Ex. 2 Solution Write the formula for area of a sector. Substitute known values. Use a calculator.  11.17 • So, the area of the sector is about 11.17 square feet.

  21. Ex. 3: Finding the Area of a Sector • A and B are two points on a P with radius 9 inches and mAPB = 60°. Find the areas of the sectors formed by APB. 60°

  22. Homework • Page 578 # 7 – 16, 20-21 & 43-45

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