Equal areas: square side to circle radius A circle and a square have the same area. What is the ratio of the side of the square to the radius of the circle?

Introduction / Context:
Geometric area equivalence allows deriving ratios among characteristic lengths. Here, a square and a circle share the same area; we are asked for the ratio of the square’s side to the circle’s radius.

Given Data / Assumptions:

• Square side = s.
• Circle radius = r.
• Areas are equal.

Concept / Approach:
Area(square) = s^2. Area(circle) = π * r^2. Equality s^2 = π r^2 implies s = r * √π. Therefore the required ratio s : r = √π : 1.

Step-by-Step Solution:
Set s^2 = π r^2.Taking square roots (positive lengths): s = r * √π.Hence s : r = √π : 1.

Verification / Alternative check:
If r = 1, then s = √π gives areas π and π, equal as required. Any consistent scaling preserves the ratio √π : 1.

Why Other Options Are Wrong:
1 : √π inverts the ratio; 1 : π and π : 1 mix area and length scales wrongly; 2 : √π has no basis in the area relationship.

Common Pitfalls:
Confusing circumference or diameter with area formula, or inverting the requested ratio accidentally.

Final Answer:
√π : 1