25. From the circular sectors of a given perimeter, find the sector of the largest area.
25. If $2 p$ is the perimeter, $R$ is the radius, and $l$ is the arc of the sector, then $2 p=2 R+l$, where $l=R \alpha, S=\frac{\pi R^{2}}{2 \pi} \alpha=R(p-R)$. Since the sum $R+(p-R)=p$ is constant, the sector has the maximum area when $R=\frac{p}{2}$, then $S=\frac{1}{4} p^{2}$.