10-7 Areas of Circles and Sectors

Quick Review

The area of a circle is A = π r 2 . eh equals , pi , r squared , .

A sector of a circle is a region bounded by two radii and their intercepted arc. The area of sector A P B = m A B ⌢ 360 ⋅ π r 2 eh p b equals . fraction m , modified eh b with frown above , over 360 end fraction . dot pi , r squared

A segment of a circle is the part bounded by an arc and the segment joining its endpoints.

Example

What is the area of the shaded region?

Area = m A B ⌢ 360 ⋅ π r 2 Use the area formula.   = 120 360 ⋅ π ( 4 ) 2 Substitute.   = 16 π 3 Simplify. The area of the shaded region is  16 π 3  ft 2 . table with 2 rows and 1 column , row1 column 1 , table with 3 rows and 3 columns , row1 column 1 , cap area , column 2 equals . fraction m , modified eh b with frown above , over 360 end fraction . dot pi , r squared , column 3 cap use the area formula. , row2 column 1 , , column 2 equals , 120 over 360 , dot pi . open 4 close squared , column 3 cap substitute. , row3 column 1 , , column 2 equals , fraction 16 pi , over 3 end fraction , column 3 cap simplify. , end table , row2 column 1 , cap the area of the shaded region is . fraction 16 pi , over 3 end fraction . ft squared , . , end table

Exercises

What is the area of each circle? Leave your answer in terms of π . pi .

Find the area of each shaded region. Round your answer to the nearest tenth.

45. 
46. 
47. A circle has a radius of 20 cm. What is the area of the smaller segment of the circle formed by a 60 ° 60 degrees  arc? Round to the nearest tenth.

10-8 Geometric Probability

Quick Review

Geometric probability uses geometric figures to represent occurrences of events. You can use a segment model or an area model. Compare the part that represents favorable outcomes to the whole, which represents all outcomes.

Example

A ball hits the target at a random point. What is the probability that it lands in the shaded region?

Since 1 3 1 third  of the target is shaded, the probability that the ball hits the shaded region is 1 3 . 1 third , .

Exercises

A dart hits each dartboard at a random point. Find the probability that it lands in the shaded region.

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