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705-805 (Hard)|   Geometry|      
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bigfernhead
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If the area of a square equals the area of a circle, which of the foll Updated on: 02 Oct 2014, 03:46
Originally posted by bigfernhead on 11 May 2009, 15:24.
Last edited by Bunuel on 02 Oct 2014, 03:46, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Question Stats:

53% (02:20) correct 47% (02:28) wrong based on 439 sessions

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If the area of a square equals the area of a circle, which of the following is closest to the ratio of the diagonal of the square to the diameter of the circle?

A. 0.95
B. 1.26
C. 1.40
D. 1.57
E. 2.51

M19-21

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B
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bigtreezl
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If the area of a square equals the area of a circle, which of the foll 13 May 2009, 22:32
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say the area of the square and the circle =100

circle:
100=pir^2
r^2~ 33
r~5.7 (closer to 6 than 5)
diameter ~11.4

square
100=s^2
s=10
diagonal = 10sqrt2= ~1.4(10)= 14

ratio square to circle = 14/11.4 ~ 14/11 = 1.27
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GMAT TIGER
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If the area of a square equals the area of a circle, which of the foll 11 May 2009, 18:43
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bigfernhead
The area of a square equals that of a circle. Which of the following is closest to the ratio of the diagonal of the square to the diameter of the circle?

0.95
1.26
1.40
1.57
2.51

I feel like I'm making a simple mistake somewhere, but can't figure out where.
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s^2 = pir^2
s = r sqrt(pi)

diagnal, ds = s (sqrt 2)
diameter, dc = 2r

ds/dc = s (sqrt 2)/2r
ds/dc = r sqrt(pi) (sqrt 2)/2r
ds/dc = sqrt(pi/2)
ds/dc = sqrt(3.14/2)
ds/dc = sqrt(1.57)
ds/dc = 1.26 approx..
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If the area of a square equals the area of a circle, which of the foll 14 May 2009, 02:08
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Area of square = Area of Circle

=> (Diagnonal ^2 / 2) = Pi * Diameter^2/4
=> Diagonal/Diameter= SQRT (Pi/4)
=> Diagonal/Diameter = SQRT (1.57)
=> Diagonal/Diameter= ~1.26
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If the area of a square equals the area of a circle, which of the foll 08 Jul 2011, 10:10
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what am I doing wrong ?

area of sq = d^2/2
area of circle = pi r^2

equating we get d/r = under root(pi/2) , which is the answer

but we need d/2r so i am getting an additional 1/2 ???
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If the area of a square equals the area of a circle, which of the foll 02 Oct 2014, 03:46
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bigfernhead
If the area of a square equals the area of a circle, which of the following is closest to the ratio of the diagonal of the square to the diameter of the circle?

A. 0.95
B. 1.26
C. 1.40
D. 1.57
E. 2.51

M19-21
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If \(x\) is the side of the square, then the area of the square is \(x^2\) and the diagonal of the square is \(x\sqrt{2}\). If \(d\) is the diameter of the circle then the area of the circle is \(\pi(\frac{d}{2})^2\). Because \(x^2 = \pi(\frac{d}{2})^2\), \(d = \frac{2x}{\sqrt{\pi}}\). The required ratio \(=\frac{x\sqrt{2}}{\frac{2x}{\sqrt{\pi}}} = \frac{\sqrt{\pi}}{\sqrt{2}} = \sqrt{\frac{\pi}{2}}\) or approximately \(\sqrt{1.57}\). This is slightly smaller than \(\sqrt{1.69} = 1.3\). The best answer is therefore B.

Answer: B
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If the area of a square equals the area of a circle, which of the foll 08 May 2020, 13:05
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Bunuel - how did d become 2x/√pi? Please could you elaborate?
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If the area of a square equals the area of a circle, which of the foll 08 May 2020, 13:14
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Bunuel
bigfernhead
If the area of a square equals the area of a circle, which of the following is closest to the ratio of the diagonal of the square to the diameter of the circle?

A. 0.95
B. 1.26
C. 1.40
D. 1.57
E. 2.51

M19-21

If \(x\) is the side of the square, then the area of the square is \(x^2\) and the diagonal of the square is \(x\sqrt{2}\). If \(d\) is the diameter of the circle then the area of the circle is \(\pi(\frac{d}{2})^2\). Because \(x^2 = \pi(\frac{d}{2})^2\), \(d = \frac{2x}{\sqrt{\pi}}\). The required ratio \(=\frac{x\sqrt{2}}{\frac{2x}{\sqrt{\pi}}} = \frac{\sqrt{\pi}}{\sqrt{2}} = \sqrt{\frac{\pi}{2}}\) or approximately \(\sqrt{1.57}\). This is slightly smaller than \(\sqrt{1.69} = 1.3\). The best answer is therefore B.

Answer: B
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\(x^2 = \pi(\frac{d}{2})^2\);

Take the square root to get \(x= \sqrt{\pi}*\frac{d}{2}\);

\(2x=\sqrt{\pi}*d\);

\(d = \frac{2x}{\sqrt{\pi}}\).
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If the area of a square equals the area of a circle, which of the foll 12 May 2023, 03:46
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