SOLUTION: The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your ans

Algebra ->  Surface-area -> SOLUTION: The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your ans      Log On


   

Question 217496: The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of π.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
perimeter of square equals 4%2As
perimeter of circle = 2%2Api%2Ar
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we have 4%2As+=+2%2Api%2Ar since the perimeters of both the square and the circle are equal.
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this makes s+=+%282%2Api%2Ar%29%2F4
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area of square equals s%5E2+=+%28%282%2Api%2Ar%29%2F4%29%5E2+=+%284%2Api%5E2%2Ar%5E2%29%2F16
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area of circle equals pi%2Ar%5E2
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ratio of area of square to area of circle equals %284%2Api%5E2%2Ar%5E2%29%2F%2816%2Api%2Ar%5E2%29
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we simplify this to get:
pi%2F4
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area of square / area of circle = pi%2F4
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this means area of square = area of circle * %28pi%2F4%29
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this also means area of circle = area of square * %284%2Fpi%29
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to test, we need a circle with a perimeter the same as a square.
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let a side of our square = 7
perimeter of the square is 4+%2A+7+=+28
perimeter of our circle must be 28
this means the radius of our circle equals 28%2F%282%2Api%29+=+4.456338407
this means the area of our circle equals pi%2Ar%5E2+=+62.38873769
the area of our square is 7%5E2+=+49
49+%2A+4%2Fpi+=+62.38873769
62.38873769+%2A+pi%2F4+=+49
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the ratio holds.
the answer is that the ratio of the area of the square to the area of the circle is pi%2F4
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