Area enclosed by two chords and a circular arc ($A=R^2(\theta +\sin \theta )$)

Take a circle of radius $R$, a vertex $A$ on its circumference, and two chords from $A$ symmetric about the diameter through $A$. If their angle is $0\le \theta \le \pi$, choose the arc between their other endpoints that does not contain $A$. The area bounded by the two chords and that arc is $R^2(\theta +\sin \theta )$. To calculate it, set $A=(0,R)$ and the other endpoints to $(\pm R\sin \theta ,-R\cos \theta )$. Their triangle has area $R^2\sin \theta (1+\cos \theta )$, and the adjoining circular segment, of central angle $2\theta$, has area $R^2(\theta -\sin \theta \cos \theta )$. The sum gives the formula, including the limiting zero-area and full-disk cases.