Squares and circles: Find the ratio of the areas of the incircle and the circumcircle of a square.

Introduction / Context:
For a square, the incircle touches sides while the circumcircle passes through vertices. Their radii relate to the square side a, enabling an area ratio.

Given Data / Assumptions:

• Square side = a
• Incircle radius r_in = a/2
• Circumcircle radius r_out = a/√2

Concept / Approach:
Area of a circle is πr^2. Ratio = (πr_in^2) : (πr_out^2) = r_in^2 : r_out^2.

Step-by-Step Solution:

Verification / Alternative check:
Set a = 2 to visualize: r_in = 1, r_out = √2; areas π and 2π give 1:2 directly.

Why Other Options Are Wrong:
1:1, 1:3, 1:4, 2:3 do not arise from the algebraic ratio of squared radii for a square.

Common Pitfalls:
Mixing diameter and radius or forgetting area depends on r^2, not r.

Final Answer:
1:2