Can you find a shape from an arbitrary Area and Perimeter?

Question

I was trying to find a new "Squircle" shape in a different way. It is a given the area and perimeter of a unit circle are $\pi$ and $2\pi$ respectively, and the area and perimeter of the circumscribed square are $4$ and $8$ respectively. Thus, the area and perimeter of the new squircle would be $\frac{\pi}{2}+4$ and $\pi+4$ respectively? Does such a shape exist? Could the methods used to find what it looks like be used for the general case of any arbitrary area and perimeter?

Answer 1

Note an area and perimeter are not enough to pin down an exact shape in general. For instance, a $3:4:5$ triangle and a rectangle of sides $3+\sqrt{3},3-\sqrt{3}$ both have an area of $6$ and perimeter of $12$.

As for the squircle, there are different kinds. One kind is the Fernández-Guasti squircle, which has two parameters, one controlling its size and one controlling its squareness. Since it has two degrees of freedom, it offers a way to find a squircle of given area and perimeter. I haven't checked what parameters would work for your particular example, but this is a starting point you can look into.