Intuitively it makes sense that the basic shape of inner region to be a square with rounded corners. I’ll leave it to the reader to try to prove this.

We need to know what the radius of the rounded corners needs to be.

We can do this by considering a circle, letting the radius vary, and finding when the value of (perimeter – area) is at a maximum.

Perimeter = 2*pi*r

Area = pi*r*r

Perimeter – Area = r*(2-r)*pi

As pi is a constant, we just need the value of r where r*(2-r) is at a maximum. Clearly this is achieved when r=(2-r), so when r=1.

So now we need to find a square with radius=1 rounded corners, whose area is equal to its perimeter. The best way is to split the shape into regions:

The four quarter circles combine to form an entire circle, and then we have an inner square and four rectangles.

Area = pi + 4x + x^2

Perimeter = 2*pi + 4x

Making these equal tells us that x^2 = pi, so x = sqrt(pi).

The side length of the bounding square is therefore sqrt(pi)+2 = 3.772…