10.150. Determine the area of the circular ring enclosed between two concentric circles, the lengths of which are $C_{1}$ and $C_{2}\left(C_{1}>C_{2}\right)$.
## Solution.

The circumference of the circles $C_{1}=2 \pi R_{1}, C_{2}=2 \pi R_{2}$, hence $R_{1}=\frac{C_{1}}{2 \pi}, R_{2}=\frac{C_{2}}{2 \pi}$. The area of the larger circle $S_{1}=\pi R_{1}^{2}=\frac{\pi C_{1}^{2}}{4 \pi^{2}}=\frac{C_{1}^{2}}{4 \pi} ; S_{2}=\frac{C_{2}^{2}}{4 \pi}$. The area of the ring $S=S_{1}-S_{2}=\frac{1}{4 \pi}\left(C_{1}^{2}-C_{2}^{2}\right)$.

Answer: $\frac{C_{1}^{2}-C_{2}^{2}}{4 \pi}$.