Equal-area circle vs. square — relate side to radius: A circle and a square have the same area. If the square has side s and the circle has radius r, determine the exact ratio s : r.

Introduction / Context:
Comparing areas of basic shapes is a staple aptitude skill. Here, a circle and a square have the same area; we must express the square’s side in terms of the circle’s radius and simplify to a clean ratio.

Given Data / Assumptions:

• Square side = s
• Circle radius = r
• Areas are equal
• Use standard formulae with π as the circle constant

Concept / Approach:
Area(square) = s^2. Area(circle) = π * r^2. If the areas are equal, set s^2 = π * r^2 and solve s/r. Reduce to simplest radical form and then write the ratio s : r.

Step-by-Step Solution:
s^2 = π * r^2Take positive roots for lengths: s = r * √πTherefore, s : r = √π : 1

Verification / Alternative check:
If r = 1, the circle’s area is π. A square with side √π has area (√π)^2 = π, matching perfectly, so √π : 1 is consistent.

Why Other Options Are Wrong:
1 : √π inverts the ratio; 1 : π and π : 1 are dimensionally incorrect for this equality; √2 : 1 is unrelated to equating s^2 and π r^2.

Common Pitfalls:
Mistaking diameter for radius; forgetting that the square’s area depends on side squared; mishandling radicals when taking square roots.

Final Answer:
√π : 1