The annulus

The area of an annulus can be found as the subtraction of the area of the biggest circle (radius $R$) minus the minor circle (radius $r$).

$$A_{annulus}=A_{big \ circle}-A_{small \ circle}=\pi \cdot R^2- \pi \cdot r^2= \pi \cdot(R^2-r^2)$$

Using proportions we will find the area of a circular trapezoid, like the one in blue in the following figure:

$$\frac{A_{circular \ trapezoid}}{A_{annulus}}=\frac{\beta_{degrees}}{360^\circ}$$

To find the area of a circular segment like the one that can be seen in the next figure, it is enough to subtract the area of the triangle $AOB$ and the area of the annulus.