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# Quick facts for circle inscribed in a square and vice versa.

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Senior Manager
Joined: 17 Mar 2011
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Quick facts for circle inscribed in a square and vice versa. [#permalink]

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20 Apr 2011, 09:47
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I often spend way too much time re-solving for these, each time I encounter them.

Circle Inscribed In A Square
The circle will have radius r, and consequentially the square will have side 2r.
What this means, is:
Ratio of area of circle to area of square = (pi)r^2 to 4r^2, or (pi)/4
Ratio of perimeter of circle to perimeter of square = 2(pi)r to 8r, or (pi)/4
Conveniently, the ratios are both the same! :D

The difference in area is (4-pi)*r^2
The difference in perimeter is (8-2pi)*r

Square Inscribed in a circle
This one is harder to solve for, since you have to do some calculations to solve for either the radius or the side of a square.
Assuming the circle has radius r, and the square will have side r*sqrt(2).
What this means, is:
Ratio of area of circle to area of square = (pi)r^2 to 2r^2, or (pi)/2
Ratio of perimeter of circle to perimeter of square = 2(pi)r to 4r*sqrt(2), or (pi)/(2*sqrt(2))

The difference in area is (pi-2)*r^2
The difference in perimeter is (2pi - 4*sqrt(2))*r
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Re: Quick facts for circle inscribed in a square and vice versa. [#permalink]

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04 Jan 2015, 12:59
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Quick facts for circle inscribed in a square and vice versa.   [#permalink] 04 Jan 2015, 12:59
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