11.4 CIRCUMFERENCE AND ARC LENGTH 11.5 AREA OF A CIRCLE & SECTORS
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1 11.4 CIRCUMFERENCE AND ARC LENGTH 11.5 AREA OF A CIRCLE & SECTORS
2 Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248 Measuring Angles The measure of an angle is determined by the amount of Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 3 rotation from the initial side to the terminal side. There are two common ways to measure angles, in degrees and in radians. degrees, denoted by the symbol º. One degree (1º) is equivalent to a rotation of 1 of one revolution. 360
3 Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian is Section the measure 4.1, Figure of a 4.5, central Illustration angle of that intercepts arc s equal in length Arc to Length, the radius pg. r 249 of the circle. In general, s r Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 6
4 Radian Measure 2 radians corresponds to 360 radians corresponds to 180 radians corresponds to
5 Radian Measure Section 4.1, Figure 4.7, Common Radian Angles, pg
6 Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by
7 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.
8 THEOREM 11.6: CIRCUMFERENCE OF A CIRCLE The circumference C of a circle is C = d or C = 2 r diameter d where d is the diameter of the circle and r is the radius of the circle.
9 EX. 1: USING CIRCUMFERENCE Find (a) the circumference of a circle with radius 6 centimeters (b) the radius of a circle with circumference 31 meters. Round decimal answers to two decimal places.
10 SOLUTION: a. b. C = 2 r = 2 6 = So, the circumference is about cm. C = 2 r 31 = 2 r 31 = r r So, the radius is about 4.93 cm.
11 ARC LENGTH 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 arc length (in linear units).
12 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 2 r AB = or Arc length of = m AB AB 360 m r P A B
13 MORE... The length of a semicircle is half the circumference, and the length of a 90 arc is one quarter of the circumference. r r ½ 2 r ¼ 2 r
14 EX. 2: FINDING ARC LENGTHS Find the length of each arc. a. 5 cm 50 A B b. C 7 cm c. 7 cm E D F
15 EX. 2: FINDING ARC LENGTHS Find the length of each arc. a. 5 cm 50 A B a. Arc length of AB = # of a. Arc length of AB = r 2 (5) 4.36 centimeters
16 EX. 2: FINDING ARC LENGTHS Find the length of each arc. b. b. Arc length of = C 7 cm D 50 CD # of 360 b. Arc length of CD 50 = r 2 (7) 6.11 centimeters
17 EX. 2: FINDING ARC LENGTHS Find the length of each arc. E c. c. Arc length of EF = 7 cm 100 F # of c. Arc length of EF = r 2 (7) centimeters
18 EX. 3: USING ARC LENGTHS Find the indicated measure. a. circumference Arc length of PQ 2 r = m PQ 360 R 60 Q P 3.82 m = 2 r 360 = 1 2 r (6) = 2 r = 2 r C = 2 r; so using substitution, C = meters.
19 EX. 3: USING ARC LENGTHS Find the indicated measure. b. m XY Arc length of X 18 in. m 2 r = XY 360 XY 18 m XY 360 = 2 (7.64) XY Z 7.64 in. Y 135 m So the m XY 135
20 EX. 4: COMPARING CIRCUMFERENCES Tire Revolutions: Tires from two different automobiles are shown on the next slide. How many revolutions does each tire make while traveling 100 feet? Round decimal answers to one decimal place.
21 EX. 4: COMPARING CIRCUMFERENCES Reminder: C = d or 2 r. Tire A has a diameter of (5.1), or 24.2 inches. Its circumference is (24.2), or about inches.
22 EX. 4: COMPARING CIRCUMFERENCES Reminder: C = d or 2 r. Tire B has a diameter of (5.25), or 25.5 inches. Its circumference is (25.5), or about inches.
23 EX. 4: COMPARING CIRCUMFERENCES Divide the distance traveled by the tire circumference to find the number of revolutions made. First, convert 100 feet to 1200 inches. TIRE A: 100 ft in in. = in. TIRE B: 100 ft in in. = in revolutions 15.0 revolutions
24 AREAS OF CIRCLES AND SECTORS The diagrams on the next slide show regular polygons inscribed in circles with radius r. Notice that as the number of sides increases, the area of the polygon approaches the value r 2.
25 EXAMPLES OF REGULAR POLYGONS INSCRIBED IN CIRCLES. 3 -gon 4 -gon 5 -gon 6 -gon
26 THEOREM. 11.7: AREA OF A CIRCLE The area of a circle is times the square of the radius or A = r 2.
27 EX. 1: USING THE AREA OF A CIRCLE Find the area of P. Solution: 8 in. P a. Use r = 8 in the area formula. A = r 2 = 8 2 = So, the area if 64, or about square inches.
28 EX. 2: USING THE AREA OF A CIRCLE Find the diameter of Z. Z Solution: b. Area of circle Z is 96 cm 2. A = r 2 96= r 2 96= r r r The diameter of the circle is about cm.
29 CIRCLE SECTOR A sector of a circle is a region bounded by two radii of the circle and their intercepted arc.
30 THEOREM 11.8: 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.
31 EX. 3: FINDING THE AREA OF A SECTOR Find the area of the sector shown below. C Sector CPD intercepts an arc whose measure is 80. The radius is 4 ft. P 4 ft. A m = r D
32 EX. 3 SOLUTION A m = r = 80 r Write the formula for area of a sector. Substitute known values. Use a calculator. So, the area of the sector is about square feet.
33 EX. 4: FINDING THE AREA OF A SECTOR A and B are two points on P with radius 9 inches and m APB = 60. Find the areas of the sectors formed by APB. 60 FIRST draw a diagram of P and APB. Shade the sectors. LABEL point Q on the major arc. FIND the measures of the minor and major arcs.
34 EX. 4: FINDING THE AREA OF A SECTOR Because m APB = 60, m = 60 and m = = 300. Use the formula for the area of a sector. Area of small sector = 60 r = = square inches
35 EX. 4: FINDING THE AREA OF A SECTOR Because m APB = 60, m = 60 and m = = 300. Use the formula for the area of a sector. Area of large sector = 300 r = = square inches
36 USING AREAS OF CIRCLES AND REGIONS You may need to divide a figure into different regions to find its area. The regions may be polygons, circles, or sectors. To find the area of the entire figure, add or subtract the areas of separate regions as appropriate.
37 EX. 5: FINDING THE AREA OF A REGION Woodworking. You are cutting the front face of a clock out of wood, as shown in the diagram. What is the area of the front of the case?
38 EX. 4: FINDING THE AREA OF A REGION
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