SOLUTION: A circle has a sector with area 17pi/2 and central angle of 17pi/9 radians. what is the area of the circle?

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Question 1136587: A circle has a sector with area 17pi/2 and central angle of 17pi/9 radians. what is the area of the circle?
Found 3 solutions by josmiceli, ikleyn, math_helper:
Answer by josmiceli(19441) About Me  (Show Source):
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The whole circle has +2pi+=+%2818pi%29+%2F9+ radians
+%28%28+17pi+%29%2F9+%29+%2F+%28%28+18pi+%29+%2F+9+%29+=+17%2F18+
Let +A+ = the area of the circle
+%28%28+17pi+%29%2F2+%29+%2F+A+=+17%2F18+
+A+=+%28+18pi+%29%2F2+
+A+=+9pi+

Answer by ikleyn(52879) About Me  (Show Source):
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.
The shortest and most straightforward way to solve this problem is to use the proportion


    sector_area%2Fcircle_area = central_angle%2F%282%2Api%29,   or


    %28%2817pi%2F2%29%29%2Fcircle_area = %28%2817pi%2F9%29%29%2F%282%2Api%29.


Or, canceling common factors 17, pi and 2 in both sides


    1%2Fcircle_area = 1%2F%289%2Api%29,   


which implies


    Circle area = 9%2Api.     ANSWER

Solved.


Answer by math_helper(2461) About Me  (Show Source):
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The ratio of the circle's area to the sector's area is equal to the ratio of the number of radians in a full circle to that of the sector:

A_circle / Area_sector = +%282%2Api%29+%2F%28%2817%2Api%2F9%29%29+
+A_circle+%2F+%28%2817%2Api%2F2%29%29+=+%282%2Api%29+%2F%28%2817%2Api%2F9%29%29+

+A_circle+=+%28%2817%2Api%2F2%29%29+%2A+%282%2Api%29+%2F%28%2817%2Api%2F9%29%29+


+A_circle+=+highlight%289+%2Api+%29+


Check:
Does this seem to make sense? Qualitatively, yes. For the sector, 17pi/2 = 8.5pi and the central angle is pi/9 less than 2pi, hence the circle should have slightly greater area than the sector, and 9pi is just a bit larger than 8.5pi.