Area of annulus from single number

An annulus is a geometric object made up of two circles like this:

The area of the annulus is clearly \( \pi R^2 - \pi r^2=\pi(R^2-r^2) \) . At first glance we might think it is enough to know the difference \( R-r \) to calculate the area of the annulus. But, since \( f(x)=x^2 \) is not a linear function this will not work. However, if we store the length of tangent of the inner circle contained in the outer circle, \( t \) , instead, the area can indeed be deduced from it:

Using pythagoras we can observe that

\[ R^2=(t/2)^2+r^2\rightarrow t^2/4=R^2-r^2 \]

and so we can calculate the area as \[ \pi t^2/4 \] .