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 \[ \pi t^2/4\].