Solution: Using $\tan(60^\circ) = \frac{\text{height}}{\text{shadow length}}$, we substitute $\tan(60^\circ) = \sqrt{3}$ and shadow length $5\,\text{cm}$. Solving for height: $\text{height} = 5 \cdot \sqrt{3}$. Thus, $\boxed{5\sqrt{3}}$.