Dear community, this is my first post on mathhelpforum and I would be really very grateful for some assistance with this problem. I am going quite crazy!

It concerns a simple 2D shape which is made of 2 intersecting circles of equal radius and distance of centres separation . I have attached a diagram of this shape as a PNG.

When angle radians, the shape is a single circle, because the two circles intersect completely. When radians, the shape is two independent circles which are just touching each other.

I have come up with expressions for the total area , and total circumference of this object:

Where

So, given radius , and the distance separating the centres of the circles, using the above expressions, you can easily calculate the total area , and the circumference .

I however need to solve theinverse problem. That is, I know the total circumference , and the total area , and I instead would like to calculate value pairs of radius and separation distance which fulfill these constraints.

Any ideas on an analytical or numerical solution very welcome!

Thanks!