Square — What is the ratio of the areas of the circumcircle and the incircle of a square?

Introduction / Context:
For a square of side a, the incircle fits inside touching the sides, and the circumcircle passes through all four vertices. Their radii relate directly to a: r_in = a/2 and r_out = a/√2. Areas scale with the square of the radius.

Given Data / Assumptions:

• Incircle radius r_in = a/2
• Circumcircle radius r_out = a/√2
• Area of a circle ∝ r^2

Concept / Approach:
The required ratio is (π r_out^2) : (π r_in^2) = r_out^2 : r_in^2.

Step-by-Step Solution:
r_out^2 = (a/√2)^2 = a^2/2r_in^2 = (a/2)^2 = a^2/4Ratio = (a^2/2) : (a^2/4) = (1/2) : (1/4) = 2 : 1

Verification / Alternative check:
Choose a = 2: r_out = √2, r_in = 1 ⇒ areas π(2) and π(1) ⇒ ratio 2:1.

Why Other Options Are Wrong:
1:2 inverts the relation; √2:1 and 1:√2 compare radii, not areas; 4:1 overstates the area ratio.

Common Pitfalls:
Confusing the ratio of radii (which is √2:1) with the ratio of areas (square of that), which is 2:1.

Final Answer:
2 : 1