The sum of perimeters of an equilateral traingle and a square is .

Area of the square is .

Area of the equilateral traingle is .

Let us consider side of the equilateral triangle that produces minimum area is .

Let us consider side of the square that produces minimum area is .

Total area : .

Sum of perimeters of an equilateral traingle and a square is .

Substitute in .

Find the minimum area.

Differentiate on each side with respect to .

Evaluate .

The dimensions of the triangle minimum area is  .

Substitute  in .

.

The dimensions of the square minimum area is .

The dimensions of the triangle for minimum area is  .

The dimensions of the square for minimum area is .