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

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

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√2)r]^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