1 / 17

7.2 Sectors of a Circles

7.2 Sectors of a Circles. Objective To find the arc length and area of a sector of a circle and to solve problem involving apparent size. Sectors of Circles. A sector of a circle, shaded at right below, is the region bounded by a central angle and the intercepted arc.

uriah-hale
Download Presentation

7.2 Sectors of a Circles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 7.2 Sectors of a Circles Objective To find the arc length and area of a sector of a circle and to solve problem involving apparent size.

  2. Sectors of Circles A sector of a circle, shaded at right below, is the region bounded by a central angle and the intercepted arc. Example 1: The radius of a pizza is 6 cm. mROQ = 60o. Find the area of the slice OQR. R 6 cm 60° O Q

  3. Sectors of Circles Example 2: The radius of a cake is 7 cm. mROP = 130o. Find the area of the slices OPR. R 7 cm O 130° P

  4. Arc Length and Area of a Sector of a Circle In general, the following formulas for the arc length s and area K of a sector with central angle  . If  is in degrees, then the arc length and the area of a sector is: If  is in radians, then the arc length and the area of a sector is: How to get the formula of area of a sector?

  5. Area of a Sector of a Circle The formula for the area of a sector of a circle can be derived from cutting the sector into n smaller sectors of the same vertices. The area of each smaller sector can be viewed as a slim and tall triangle with height r and base si . The area of a sector then is the sum of area of all those n smaller triangles.  r s1 sn s2 s3 r The sum of all n bases for those smaller triangle is actually the length of the arc, which is r . Therefore,

  6. Area of a Circle Hence, the formula for the area of a a circle is:   r or

  7. Area of a Sector of a Circle The formula for the area of a sector of a circle is:   r Again  must be in RADIANS so if it is in degrees you must convert to radians to use the formula. Find the area of the sector if the radius is 3 feet and  = 50° = 0.873 radians

  8. Your Turn!

  9. Apparent Size When there is nothing in our field of vision against which to judge the size of an object, we perceive the object to be smaller when it is farther away. For example, the sun is much larger than the moon, but we perceive the sun to be about the same size as the moon because the sun is so much farther from Earth. So, how big an object looks depends not only on its size but also on the angle that it subtends at our eyes. The measure of this angle is called the object’s apparent size.

  10. Example 2 Jupiter has an apparent size of 0.01o when it is 8 x 108 km from Earch. Find the approximate diameter of Jupiter. As the exaggerated diagram above indicates, the diameter of Jupiter is approximately the same as the arc length of a sector with central angle 0.01o and radius 8 x 108 km.

  11. Example 3 A phonograph record with diameter 12 in., turns at 331/3 rpm. Find the distance that a point on the rim travels in one minute. 6 in

  12. Example 4 A sector has perimeter 16 cm and area 15 cm2. Find its radius r and arc length s.

  13. A Useful Concept A SEGMENT is a region bounded by a chord and its intercepted arc A segment is a minor segment if the intercepted arc is less than 180 degrees Area of minor segment = (Area of sector) – (Area of triangle)

  14. Example 5 The radius of a circle is 12 yd. mROQ = 90o. Find the area of segment RQ . R Area of minor segment = (Area of sector) – (Area of triangle) O Q 12 yd

  15. Example 6: Find the area of the shaded region. r2 4 cm (4)2 16 16 - 123 4 cm 2 30o  29.5 cm2 23 43

  16. AssignmentP. 264 #1 – 10, 11, 13, 17

More Related