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.

Question

Philoid · Accepted Answer

Let us say the sum of perimeter of square and circumference of circle be L

Given: Sum of the perimeters of a square and a circle.

Assuming, side of square = a and radius of circle = r

Then, L = 4a + 2πr …1

Let the sum of area of square and circle be S

So, S = a² + πr²

S =

Condition for maxima and minima

= 0

So, for >0

This is the condition for minima

From equation 1

Substituting from equation 2

a = 2r

Hence, proved.