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 $r$).

$$A_{annulus}=A_{bigcircle}-A_{smallcircle}=\pi \cdot R^2-\pi \cdot r^2=\pi \cdot (R^2-r^2)$$

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

$$\frac{A_{circulartrapezoid}}{A_{annulus}}=\frac{{\beta }_{degrees}}{{360}^{\circ }}$$

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.