# Areas of Circles and Sectors

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Areas of Circles and Sectors
Lesson 10-7 1. A park contains two circular playgrounds. One has a diameter of 60 m, and the other has a diameter of 40 m. How much greater is the area of the larger playground? Round to the nearest whole number. 2. A circle has an 8-in. radius. Find the area of a sector whose arc measures 135. Leave your answer in terms of . For Exercises 3 and 4, find the area of the shaded segment. Round to the nearest whole unit. Lesson Quiz 1571 m2 24 in.2 15 cm2 138 in.2 10-8

Geometric Probability
Lesson 10-8 Check Skills You’ll Need (For help, go to the Skills Handbook, pages 756 and 762.) 4. Two circles have radii 1 m and 2 m, respectively. What is the simplest form of the fraction with numerator equal to the area of the smaller circle and denominator equal to the area of the larger circle? an odd number 7. 2 or a prime number BD AE AB BC CE AF Find and simplify each ratio. You roll a number cube. Find the probability of rolling each of the following. The area of the smaller circle is r2 = (1)2 =  m; The area of the larger circle is r2 = (2)2 = 4 m; smaller : larger =  : 4 = 1 : 4, or ¼. Check Skills You’ll Need 10-8

Geometric Probability
Lesson 10-8 Notes You may recall that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes. 10-8

Geometric Probability
Lesson 10-8 Notes Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure. 10-8

Geometric Probability
Lesson 10-8 Notes For example, if points of segments represent outcomes, then 10-8

Geometric Probability
Lesson 10-8 Notes If an event has a probability p of occurring, the probability of the event not occurring is 1 – p. Remember! 10-8

Geometric Probability
Lesson 10-8 Additional Examples Finding Probability Using Segments A gnat lands at random on the edge of the ruler below. Find the probability that the gnat lands on a point between 2 and 10. The length of the segment between 2 and 10 is 10 – 2 = 8. The length of the ruler is 12. P(landing between 2 and 10) = = , or length of favorable segment length of entire segment 8 12 2 3 Quick Check 10-8

Geometric Probability
Lesson 10-8 Additional Examples Real-World Connection A museum offers a tour every hour. If Benny arrives at the tour site at a random time, what is the probability that he will have to wait at least 15 minutes? Quick Check Because the favorable time is given in minutes, write 1 hour as 60 minutes. Benny may have to wait anywhere between 0 minutes and 60 minutes. Represent this using a segment. Starting at 60 minutes, go back 15 minutes. The segment of length 45 represents Benny’s waiting more than 15 minutes. P(waiting more than 15 minutes) = , or 45 60 3 4 The probability that Benny will have to wait at least 15 minutes is , or 75%. 3 4 10-8

Geometric Probability
Lesson 10-8 Notes If the points of a region represent equally-likely outcomes, then you can find probabilities by comparing areas. 10-8

Geometric Probability
Lesson 10-8 Additional Examples Finding Probability Using Area A circle is inscribed in a square target with 20-cm sides. Find the probability that a dart landing randomly within the square does not land within the circle. Find the area of the square. A = s2 = 202 = 400 cm2 Find the area of the circle. Because the square has sides of length 20 cm, the circle’s diameter is 20 cm, so its radius is 10 cm. A = r 2 = (10)2 = cm2 Find the area of the region between the square and the circle. A = (400 – ) cm2 10-8

Geometric Probability
Lesson 10-8 Additional Examples (continued) Use areas to calculate the probability that a dart landing randomly in the square does not land within the circle. Use a calculator. Round to the nearest thousandth. P (between square and circle) = = = 1 – area between square and circle area of square 400 – 100 400 4 The probability that a dart landing randomly in the square does not land within the circle is about 21.5%. Quick Check 10-8

Geometric Probability
Lesson 10-8 Additional Examples Real-World Connection To win a prize, you must toss a quarter so that it lands entirely within the outer region of the circle below. Find the probability that this happens with a quarter of radius in. Assume that the quarter is equally likely to land anywhere completely inside the large circle. 15 32 The center of a quarter with a radius of in. must land at least in. beyond the boundary of the inner circle in order to lie entirely outside the inner circle. Because the inner circle has a radius of 9 in., the quarter must land outside the circle whose radius is 9 in in., or in. 15 32 Remove this slide. Do not replace. 10-8

Geometric Probability
Lesson 10-8 Additional Examples (continued) 15 32 Find the area of the circle with a radius of in. A = r2 = (9 ) in.2 15 32 15 32 Similarly, the center of a quarter with a radius of in. must land at least in. within the outer circle. Because the outer circle has a radius of 12 in., the quarter must land inside the circle whose radius is 12 in. – in., or in. 17 Remove this slide. Do not replace. Find the area of the circle with a radius of in. A = r2 = ( ) in.2 17 32 10-8

Geometric Probability
Lesson 10-8 Additional Examples (continued) Use the area of the outer region to find the probability that the quarter lands entirely within the outer region of the circle. P (outer region) = area of outer region area of large circle = The probability that the quarter lands entirely within the outer region of the circle is about 0.326, or 32.6%. Quick Check 10-8

Geometric Probability
Lesson 10-8 Lesson Quiz 1. A point on AF is chosen at random. What is the probability that it is a point on BE? 2. Express elevators to the top of a tall building leave the ground floor every 40 seconds. What is the probability that a person would have to wait more than 30 seconds for an express elevator? A dart you throw is equally likely to land at any point on each board shown. For Exercises 3–5, find the probability of its landing in the shaded area. 3. regular octagon 4. square 5. circle 3 5 1 4 3 8 , or 37.5% 1 2 , or 50% 8 25 , or 32% 10-8

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