A sector of a circle has a central angle of $150^{\circ}$. Find the area of the sector if the radius of the circle is $10 \mathrm{~cm}$.

Step 1 ：Let's denote the area of the sector as $A$ cm$^2$.

Step 2 ：The area of a sector is given by the formula $A = \frac{r^2 \cdot \theta}{2}$, where $r$ is the radius and $\theta$ is the central angle in radians.

Step 3 ：First, we need to convert the central angle from degrees to radians. We know that $180^\circ = \pi$ radians, so $150^\circ = \frac{150}{180} \cdot \pi = \frac{5}{6} \cdot \pi$ radians.

Step 4 ：Substitute $r = 10$ cm and $\theta = \frac{5}{6} \cdot \pi$ into the formula, we get $A = \frac{10^2 \cdot \frac{5}{6} \cdot \pi}{2} = 25 \cdot \pi$ cm$^2$.

Step 5 ：So, the area of the sector is $\boxed{25 \pi}$ cm$^2$.