I am trying to come up with a way to calculate the cross-sectional area of the shape shown in the figure below. My first method would be to subtract the circle from the rectangle like this: $$(Y)\left(\frac{OD-ID}{2}\right)-A_{circle}$$, however, I do not know how to calculate the area of this circle since the OD and ID are not tangent to it. Also, I would prefer to use the $'X'$ dimension since $'Y'$ is a reference only.
You’re looking for the area of a circular segment of radius $R$ and chord length $C={OD-ID\over2}$. This can be found by subtracting the area of an isosceles triangle from the area of a sector of a circle: $$A=\frac12R^2(\theta-\sin\theta),$$ where the angle $\theta$ can be found via the relationship $C=2R\sin{\frac\theta2}$.