Question

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

Answer 1

Let side of square is x and radius of circle is r.

Perimeter of circle (circumference) = 2πr

Sum of both perimeters = 4x + 2πr = k …..(i)

Area of circle, A₂ = πr²

Area of square, A₂ = x²

∴ Sum of areas, A = πr² + x²

Hence, side of square is equal to diameter of circle, when sum of area of both is minimum.