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

Question

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

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

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

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

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

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

Archit3110 · Answer

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

baru · Answer

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

EgmatQuantExpert · Answer

Given

• A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square.

To Find

• The ratio of the area of the smaller circle to the area of the larger square.

Approach and Working Out

• In the image AB = 2r (Say)
• AB = side of the smaller square
 o CD = diagonal of the smaller square
o CD = 2  \sqrt{2 } r 
• EF = CD = Diameter of the bigger circle.
 o EF = 2  \sqrt{2 } r 
• Area of the smaller circle = π  r^2 
• Area of the bigger square =  (2 \sqrt{2 } r) ^2  
 o = 8  r^2  
• Ratio = π : 8

Correct Answer: Option D

ScottTargetTestPrep · Answer

Solution:
 
Let the radius of the smaller circle be r. Then, the area of the smaller circle is ?r^2.

Since the smaller circle is inscribed in the smaller square, a side of the smaller square is equal to the diameter of the smaller circle, which is 2r.

Since the smaller square is inscribed in the larger circle, the diameter of the larger circle is the diagonal of the smaller square. Since the side length of the smaller square is 2r, the diameter of the larger circle (which is the diagonal of the smaller square) is 2r * √2 = (2√2)r.

Finally, since the larger circle is inscribed in the larger square, the side length of the larger square is the diameter of the larger circle, which is (2√2)r. Thus, the area of the larger square is  ^2 = 8r^2.

Thus, the ratio of the area of the smaller circle to the area of the larger square is (?r^2) / (8r^2) = ?/8.

Answer: B

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards