The areas of a circle and a square are equal. What is the square of the ratio of the length of the diameter of the circle to the length of the diagonal of the square?

Introduction / Context:
This question links the area of a circle and the area of a square, then asks about a ratio involving a diameter and a diagonal. It tests your ability to work with area formulas, connect them algebraically and then manipulate the resulting expressions to find a specific ratio. Such questions are common in geometry based aptitude tests.

Given Data / Assumptions:

• Area of the circle equals the area of the square.

• Let radius of the circle be r and side of the square be a.

• We need the square of the ratio (diameter of circle) / (diagonal of square).

• Use pi approximately equal to 22 / 7 where required for comparison with options.

Concept / Approach:
Start by equating the areas of the circle and the square. The area of a circle is pi * r^2, and the area of a square is a^2. From pi * r^2 = a^2 we can express the side a of the square in terms of r. Then we compute the diameter of the circle, which is 2r, and the diagonal of the square, which is a * sqrt(2). After that, we form the ratio, simplify it and finally square it to match the options.

Step-by-Step Solution:
Let radius of circle = r and side of square = a. Area of circle = pi * r^2. Area of square = a^2. Given areas are equal, so pi * r^2 = a^2. Thus, a = r * sqrt(pi). Diameter of circle = 2r. Diagonal of square = a * sqrt(2) = r * sqrt(pi) * sqrt(2) = r * sqrt(2 * pi). Ratio = (diameter) / (diagonal) = 2r / (r * sqrt(2 * pi)) = 2 / sqrt(2 * pi). Square of the ratio = (2 / sqrt(2 * pi))^2 = 4 / (2 * pi) = 2 / pi. Using pi ≈ 22 / 7, square of ratio ≈ 2 / (22 / 7) = 2 * (7 / 22) = 14 / 22 = 7 / 11.

Verification / Alternative check:
To verify, note that 7 / 11 is approximately 0.636. If pi is taken as 3.14, then 2 / pi ≈ 2 / 3.14 ≈ 0.6369, which is very close to 0.636. This confirms that 7 / 11 is a good rational approximation for 2 / pi, and therefore it matches the expression derived for the square of the ratio. No other given option provides such a close match to 2 / pi.

Why Other Options Are Wrong:
The value 11 / 7 is approximately 1.57, which is larger than 1 and cannot be equal to 2 / pi which is less than 1. The value 7 / 9 is about 0.78 and does not match 2 / pi. The value 9 / 7 is approximately 1.28 and is also too large. Only 7 / 11, approximately 0.636, closely matches the theoretical value of 2 / pi derived from the algebraic steps.

Common Pitfalls:
One common error is to confuse diagonal and side of the square, or to forget the factor sqrt(2) in the diagonal formula. Some students also mistakenly set diameter equal to side of the square instead of working through the area equality carefully. Another pitfall is approximating pi too early in the solution, which can hide the underlying exact relationship and make algebraic simplifications harder to see.

Final Answer:
The square of the ratio of the diameter of the circle to the diagonal of the square is 7/11.