When is the largest area between a circular arc of a given length and its chord?
For a circle of radius $r$ with a central angle $\alpha$ of the arc, $T'(\alpha)>0$ when $0<\alpha<\pi$, however, $T'(\alpha)<0$ when $\pi<\alpha<2 \pi$.

From the penultimate form, however, we see that the derivative is also defined when $\cos \frac{\alpha}{2}=0$, that is, when $\alpha=\pi$, and its value is $T'(\pi)=0$. A maximum can only occur here, and according to the sign of the derivative, it is a strict maximum in the proper sense for $T(\alpha)$.

Thus, the area in question is the largest when the given length is bent into a semicircle, and in this case its value is $T=\frac{1}{2 \pi}$.