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A circle and a square have the same area. What is the ratio

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A circle and a square have the same area. What is the ratio  [#permalink]

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New post Updated on: 16 Sep 2010, 17:03
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A circle and a square have the same area. What is the ratio of the diameter of the circle to the diagonal of the square?

(A) \(2 : \sqrt{(2\pi)}\)
(B) \(1 : 2\sqrt{\pi}\)
(C) \(2\sqrt{\pi} : \sqrt{2}\)
(D) \(1 : \sqrt{2}\)
(E) \(1 : 2\pi\)

Originally posted by zisis on 15 Sep 2010, 13:36.
Last edited by zisis on 16 Sep 2010, 17:03, edited 3 times in total.
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Re: MGMAT Challenge Problem Showdown  [#permalink]

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New post Updated on: 16 Sep 2010, 17:05
1
IMO A

diameter 20,
radius= 10
πr^2 = 100π
area = 100π
side = \(\sqrt{100\pi}\)
diagonal = \(Side * \sqrt{2} = 10\sqrt{(2\pi)}\)

ratio
\(20 : 10\sqrt{(2\pi)}\)
= \(2 : \sqrt{(2\pi)}\)

Originally posted by zisis on 15 Sep 2010, 13:41.
Last edited by zisis on 16 Sep 2010, 17:05, edited 2 times in total.
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A circle and a square have the same area. What is the ratio  [#permalink]

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New post 15 Sep 2010, 13:47
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zisis wrote:
A circle and a square have the same area. What is the ratio of the diameter of the circle to the diagonal of the square?

(A) 2 : √(2)
(B) 1 : 2√
(C) 2√ : √2
(D) 1 : √2
(E) 1 : 2


\(area_{circle}=\pi{\frac{diameter^2}{4}}=area_{square}=\frac{diagonal^2}{2}\);

\(\pi{\frac{diameter^2}{4}}=\frac{diagonal^2}{2}\);

\(\frac{diameter^2}{diagonal^2}=\frac{2}{\pi}\);

\(\frac{diameter}{diagonal}=\frac{\sqrt{2}}{\sqrt{\pi}}=\frac{2}{\sqrt{2\pi}}\).

Answer: A.

Pleas format the answers properly.
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Re: MGMAT Challenge Problem Showdown  [#permalink]

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New post 15 Sep 2010, 13:50
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zisis wrote:
A circle and a square have the same area. What is the ratio of the diameter of the circle to the diagonal of the square?

(A) 2 : √(2π)
(B) 1 : 2√π
(C) 2√π : √2
(D) 1 : √2
(E) 1 : 2π


\(\pi r^2 = a^2\)

\(\frac{r}{a} = \frac{1}{\sqrt{\pi}}\)

\(\frac{2r}{\sqrt{2}a} = \frac{2}{\sqrt{2\pi}}\)

Hence A
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Re: MGMAT Challenge Problem Showdown  [#permalink]

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New post 15 Sep 2010, 14:47
Had to copy and paste (pi). Is there a better way?

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Re: MGMAT Challenge Problem Showdown  [#permalink]

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New post 15 Sep 2010, 14:50
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A circle and a square have the same area. What is the ratio  [#permalink]

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New post 06 Jan 2015, 22:48
Answer = A

Let the area of circle = area of square = 1

\(\pi r^2 = 1\)

\(r = \frac{1}{\sqrt{\pi}}\)

2r = diameter \(= \frac{2}{\sqrt{\pi}}\)

Side of square = 1

Diagonal \(= \sqrt{2}\)

Ratio \(= \frac{2}{\sqrt{2*\pi}}\)
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Re: A circle and a square have the same area. What is the ratio  [#permalink]

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New post 20 Aug 2018, 11:04
zisis wrote:
A circle and a square have the same area. What is the ratio of the diameter of the circle to the diagonal of the square?

(A) \(2 : \sqrt{(2\pi)}\)
(B) \(1 : 2\sqrt{\pi}\)
(C) \(2\sqrt{\pi} : \sqrt{2}\)
(D) \(1 : \sqrt{2}\)
(E) \(1 : 2\pi\)


Let us denote the length of a side of the square by s and the radius of the circle by r.

We can create the equation:

πr^2 = s^2

r√π = s

Since the side of the square s = r√π, the diagonal of the square is r√π x √2 = r√(2π).

The diameter of the circle is 2r. Thus, the ratio of the diameter of the circle to the diagonal of the square is:

2r/[r√(2π)] = 2/√(2π)

Answer: A
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Re: A circle and a square have the same area. What is the ratio   [#permalink] 20 Aug 2018, 11:04
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