Math problem: Determine the corner radius of an inner border based on outer corner radius/thickness

Here's a math/geometry problem for the math whizzes (not my strongest subject). This is for WPF, but should be general e开发者_如何学运维nough to solve regardless:

I have two embedded Border elements, with the outer one having a certain corner radius, R and border thickness, T. Given these two values, what should the corner radius of the inner Border, R' be set to such that the two corner edges meet with no overlap or holes?

So far I've just been eyeballing it, but if someone can give me a proper formula that would be great. Respect points if you can!! ;)

T'/2 + R` = R - T/2

or

For a given T, R and T' then R' = R - T/2 - T'/2

So for instance for an outer border thickness of (T=10), and radius of (R=8), and an inner border thickness of (T'=4), you'd need an inner border radius of 1.