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## Warm Up

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**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.**The circumference C of a circle is C = d or C = 2r,**where d is the diameter of the circle and r is the radius of the circle. Theorem 11.6: Circumference of a Circle**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).**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 = 2r 360° m Arc length = • 2r 360°**The length of a semicircle is half the circumference, and**the length of a 90° arc is one quarter of the circumference. More . . . ½ • 2r r r ¼ • 2r**Ex. 2: Finding Arc Lengths**• Find the length of each arc. a. b. c. 50° 100° 50°**• 2(5)**a. Arc length of = 50° 360° Ex. 2: Finding Arc Lengths • Find the length of each arc. # of ° a. • 2r a. Arc length of = 360° 50° 4.36 centimeters**Ex. 2: Finding Arc Lengths**• Find the length of each arc. # of ° b. • 2r b. Arc length of = 360° 50° 50° • 2(7) b. Arc length of = 360° 6.11 centimeters**Ex. 2: Finding Arc Lengths**• Find the length of each arc. # of ° c. • 2r c. Arc length of = 360° 100° 100° • 2(7) c. Arc length of = 360° 12.22 centimeters**m**Ex. 3: Using Arc Lengths • Find the indicated measure. a. circumference Arc length of = 2r 360° 3.82 = 60° 2r 360° 60° 3.82 1 = 2r 6 3.82(6) = 2r 22.92 = 2r C = 2r; so using substitution, C = 22.92 meters.**Ex. 3: Using Arc Lengths**• Find the indicated measure. b. m Arc length of m = 2r 360° 18 m = 360° • • 360° 2(7.64) 360° 135° m**Theorem 10-11: Area of a Circle**• The area of a circle is times the square of the radius or A = r2.**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.**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.**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°**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°**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.**Ex. 3: Finding the Area of a Sector**• A and B are two points on a P with radius 9 inches and mAPB = 60°. Find the areas of the sectors formed by APB. 60°**Homework**• Page 578 # 7 – 16, 20-21 & 43-45