If we call the length of the rectangle ‘x’ and the height of the rectangle ‘y’, then the value we are seeking is 2x+2y-xy.

If we just look at one half of the figure, replacing the circle with some radii and showing tangent lines as equal we get:

We now have a right-angled triangle with sides x, y and (x+y-2).

x^2+y^2 = (x+y-2)^2

x^2+y^2 = x^2+2xy-4x+y^2-4y+4

Subtract x^2+y^2 from both sides:

0 = 2xy-4x-4y+4

Subtract 2xy-4x-4y from both sides:

4x+4y-2xy = 4

Divide all terms by 2

2x+2y-xy = 2

The value of 2x+2y-xy is what we are seeking, and so the answer is 2. There isn’t enough information to determine x and y, but the difference between the perimeter and the area will be constant.