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# A circle is inscribed in a square, then a square is inscribed in this

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Math Expert
Joined: 02 Sep 2009
Posts: 54401
A circle is inscribed in a square, then a square is inscribed in this  [#permalink]

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24 Mar 2019, 22:48
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Difficulty:

65% (hard)

Question Stats:

44% (03:12) correct 56% (02:11) wrong based on 16 sessions

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A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

(A) $$\frac{\pi}{16}$$

(B) $$\frac{\pi}{8}$$

(C) $$\frac{\pi}{316}$$

(D) $$\frac{\pi}{4}$$

(E) $$\frac{\pi}{2}$$

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Re: A circle is inscribed in a square, then a square is inscribed in this  [#permalink]

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25 Mar 2019, 00:46
1
Bunuel wrote:
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

(A) $$\frac{\pi}{16}$$

(B) $$\frac{\pi}{8}$$

(C) $$\frac{\pi}{316}$$

(D) $$\frac{\pi}{4}$$

(E) $$\frac{\pi}{2}$$

draw figure a per given description
let side of large square = 8
8 will be equal to diameter of large circle = radius = 4
and diameter of circle = diagonal of small square = 8 = s√2 ; s= 4√2
4√2 = diameter of small circle so ; radis = 2√2
area of small circle (2√2)^2 *pi = 8pi
and large square area = 8^2 = 64 so ratio
8pi/64 = pi/8 IMO B
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A circle is inscribed in a square, then a square is inscribed in this  [#permalink]

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25 Mar 2019, 23:23
1
let x be the side of the big square S1 => diagonal of S1 = diameter of inscribed circle C1 = $$\frac{x}{(\sqrt{2})}$$
side of inscribed square in CircleC1,S2 => diameter of Circle C1 =$$\frac{x}{(\sqrt{2})}$$
radius of CircleC2 inscribed in S2 ,C2 => Side of S2 =$$\frac{x}{2(\sqrt{2})}$$

Area of Small circle C2 = $$\pi(\frac{x}{2(\sqrt{2})})^2$$
Area of Square S1 = $$x^2$$

Ratio = Area of C2/ Area of S1
= $$\frac{\frac{\Pi x^2}{8}}{ x^2}$$
= $$\frac{\Pi}{8}$$ =option B
A circle is inscribed in a square, then a square is inscribed in this   [#permalink] 25 Mar 2019, 23:23
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