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# A circle is inscribed in a half circle with a diameter of

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A circle is inscribed in a half circle with a diameter of [#permalink]  23 Feb 2012, 00:53
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Question Stats:

57% (01:35) correct 43% (01:06) wrong based on 60 sessions
A circle is inscribed in a half circle with a diameter of $$\pi$$ . What is the ratio of the area of the half circle to the area not covered by the inscribed circle?

1:1
1:2
1:4
3:4
2:1

I know this has been discussed before & answer is easy to figure out. But my doubt is how to figure out the area of the inscribed circle.
[Reveal] Spoiler: OA
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Re: Circles [#permalink]  23 Feb 2012, 02:46
Expert's post
GMATPASSION wrote:
A circle is inscribed in a half circle with a diameter of $$\pi$$ . What is the ratio of the area of the half circle to the area not covered by the inscribed circle?

1:1
1:2
1:4
3:4
2:1

I know this has been discussed before & answer is easy to figure out. But my doubt is how to figure out the area of the inscribed circle.

This question needs a revision, because there are infinitely many circles possible to inscribe in a semicircle. Consider just two cases below:
Attachment:

Semicircle.PNG [ 18.1 KiB | Viewed 3463 times ]
Now, intended meaning of the question is the case with the circle which inscribed right in the middle but it should be specified.

Anyway, for that case we don't really need any formula to get the ratio:
Since the radius of the big circle ($$\frac{\pi}{2}$$) is twice the radius of the inscribed circle ($$\frac{\pi}{4}$$) then its area is 4 times greater then the area of the inscribed circle (because in the area formula the radius is squared. For example the area of a circle with radius of 2 is $$4\pi$$, which is 4 times greater than the radius of a circle with the radius of 1, which is $$\pi$$).

Thus the are of the semicircle is 4/2=2 times greater than the area of the inscribed circle: the area of the semicircle 2 units, the area of the inscribed circle 1 unit, and the area of the semicircle not covered by the inscribed circle is also 1. Ratio: 2/1.

Hope it's clear.
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Re: A circle is inscribed in a half circle with a diameter of [#permalink]  23 Feb 2012, 03:12

The inscribed circle will have diameter equal to radius of half circle....

area of unshaded region would be = area of semi-circle - area of inscribed circle
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Re: A circle is inscribed in a half circle with a diameter of [#permalink]  23 Feb 2012, 05:19
area of full circle is picube/4
area of half circle is picube /8

now area of circle with radius pi/4 is picube /16
area not covered by inscribed circle is picube /8 - picube /16 = picube /16

area of half circle / area not covered by inscribed circle = picube/8 divides by picube/16
which will give us ratio 2:1

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Re: A circle is inscribed in a half circle with a diameter of [#permalink]  25 Nov 2014, 09:11
'A circle is inscribed right in the middle of a semicircle with a diameter of 'Pi' as shown below....' is the actual question.

For me this question is ambiguous since I don't understand which circle has the diameter of 'Pi' whether it is the semicircle or the inscribed circle. BUT WHAT I LIKE ABOUT THIS QUESTION IS SINCE IT'S A RATIO QUESTION , NO MATTER FOR WHICH CIRCLE YOU CONSIDER THE DIAMETER AS 'Pi' YOU GET THE SAME ANSWER 2:1
Re: A circle is inscribed in a half circle with a diameter of   [#permalink] 25 Nov 2014, 09:11
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