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Circumference and Arc Length

Circumference and Arc Length. Created By: Mrs. Spitz 2006 Found & updated by Mrs. Tenney 2010 Ripped off by Ms. Young a few days later. Objectives/Assignment. Find the circumference of a circle and the length of a circular arc. Use circumference and arc length to solve real-life problems.

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Circumference and Arc Length

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  1. Circumference and Arc Length Created By: Mrs. Spitz 2006 Found & updated by Mrs. Tenney2010 Ripped off by Ms. Young a few days later

  2. Objectives/Assignment • Find the circumference of a circle and the length of a circular arc. • Use circumference and arc length to solve real-life problems.

  3. Finding circumference andarc 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: Circumference of a Circle

  5. Ex. 1: Using circumference • Find (a) the circumference of a circle with radius 6 centimeters and (b) the radius of a circle with circumference 31 meters. Round decimal answers to two decimal places.

  6. C = 2r = 2 •  • 6 = 12  37.70 So, the circumference is about 37.70 cm. C = 2r 31 = 2r 31 = r Divide by 2 4.93  r Solution: a. b. Given r = 6, find C Given C = 31, find r 2

  7. 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). Ex: cm, in, or ft.

  8. 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 of m = 2r 360° m or Arc length of = • 2r 360°

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

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

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

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

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

  14. Ex. 3: Using Arc Lengths • Find the indicated measure. a. circumference Arc length of # of ° = 2r 360° 3.82 60° = 2r 360° 60° 3.82 60° = Cross Multiply! C 360° 3.82(360) = 60C Divide by 60 22.92 = C C = 2r; so using substitution, C = 22.92 meters.

  15. Ex. 3: Using Arc Lengths • Find the indicated measure. b. m Arc length of # of ° = 2r 360° 18 Z° = 2r 360° 18 Z° = Cross Multiply 2(7.64) 360° 18(360) = 2 (7.64)z 6480 = 48z Divide by 48 135° = z

  16. Try the one on your paper!

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