The answer is MUCH simpler!

You didn't provide any information other than the length of the inner tangent. Ergo, the answer must be true (the same) for any pair of circles. So, take the simplest case where the inner circle goes to zero and the diameter of the big circle is 32. It's area is 16*16*pi. There's your 256!

It's rather cool, that the constant tangent length yield a constant area. Your solution proves that. Mine assumed it based on meta-evidence.