1. 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
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2. 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

6. 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.
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7. 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.
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8. Geometric Probability
Lesson 10-8
Notes
For example, if points of segments represent outcomes, then
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9. 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!
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10. 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
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11. 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
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12. Geometric Probability
Lesson 10-8
Notes
If the points of a region represent equally-likely outcomes, then you can find probabilities by comparing areas.
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13. 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
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14. 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
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15. 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.
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16. 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
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17. 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
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18. 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%
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