Find the ratio of the areas of the incircle and the circumcircle of a square.\n(Assume the square has side length s. The incircle touches all 4 sides, and the circumcircle passes through all 4 vertices.)

Introduction:
This question tests geometric relationships between a square and its associated circles. The incircle of a square is the largest circle that fits inside the square, touching all four sides. The circumcircle is the circle that passes through all four vertices of the square. The ratio of their areas depends only on their radii. Since circle area is proportional to r^2, we need to express both radii in terms of the square’s side length, then compare the squared radii. The problem checks whether you remember how to relate a square’s diagonal to the circumradius and how the incircle radius relates to the side length.

Given Data / Assumptions:

• Let square side = s
• Incircle radius r_in = s/2 (diameter equals side)
• Square diagonal = s*sqrt(2)
• Circumcircle radius r_out = (diagonal)/2 = s*sqrt(2)/2 = s/sqrt(2)
• Circle area formula: A = pi * r^2

Concept / Approach:
Compute r_in^2 and r_out^2 in terms of s, then take ratio A_in : A_out = (pi*r_in^2) : (pi*r_out^2). The pi cancels, so only squared radii matter. This makes the solution clean and independent of any numeric side length.

Step-by-Step Solution:
r_in = s/2 => r_in^2 = s^2/4r_out = s/sqrt(2) => r_out^2 = s^2/2Area ratio = r_in^2 : r_out^2 = (s^2/4) : (s^2/2)Cancel s^2: ratio = (1/4) : (1/2)Multiply both terms by 4: ratio = 1 : 2

Verification / Alternative Check:
Pick an easy value, say s = 2. Then incircle radius = 1, area = pi*1^2 = pi. Diagonal = 2*sqrt(2), circumradius = sqrt(2), area = pi*(sqrt(2)^2) = 2pi. Ratio = pi : 2pi = 1 : 2. This numeric check confirms the algebraic result.

Why Other Options Are Wrong:
1:4 or 4:1: would require one radius to be double the other, which is not true here.2:1: reverses the correct ratio.3:2: does not match the squared-radius relationship of a square’s circles.

Common Pitfalls:
Using side s as the circumcircle diameter (incorrect; diagonal is the diameter).Forgetting area depends on r^2, not r.Mixing up inradius and circumradius formulas.Not cancelling pi and s^2 cleanly, leading to confusion.

Final Answer:
1 : 2