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

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Joined: 16 Feb 2010
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A circle and a square have the same area. What is the ratio  [#permalink]

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Updated on: 16 Sep 2010, 17:03
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Difficulty:

45% (medium)

Question Stats:

72% (02:10) correct 28% (02:28) wrong based on 163 sessions

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

diameter 20,
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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15 Sep 2010, 13:47
2
1
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

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

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

$$\frac{diameter^2}{diagonal^2}=\frac{2}{\pi}$$;

$$\frac{diameter}{diagonal}=\frac{\sqrt{2}}{\sqrt{\pi}}=\frac{2}{\sqrt{2\pi}}$$.

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Re: MGMAT Challenge Problem Showdown  [#permalink]

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15 Sep 2010, 13:50
1
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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15 Sep 2010, 14:47
Had to copy and paste (pi). Is there a better way?

Posted from my mobile device
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Posts: 56304
Re: MGMAT Challenge Problem Showdown  [#permalink]

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15 Sep 2010, 14:50
zisis wrote:
Had to copy and paste (pi). Is there a better way?

Posted from my mobile device

Mark \pi by [m] button. Also check: http://gmatclub.com/forum/writing-mathe ... 72468.html

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

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06 Jan 2015, 22:48

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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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)

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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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