Given the sum of the perimeters of a square and a circle, show that the sum of their areas is least when one side of the square is equal to diameter of the circle.

Explanation

To solve the problem, we need to show that the sum of the areas of a square and a circle is minimized when the side of the square is equal to the diameter of the circle, given that the sum of their perimeters is constant.

### Step 1: Define the variables
Let:
- \( A \) = side length of the square
- \( R \) = radius of the circle

### Step 2: Write the expressions for the perimeters
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