2.23. A cone and a hemisphere have a common base, the radius of which is equal to $R$. Find the lateral surface area of the cone, if its volume is equal to the volume of the hemisphere.
2.23. Since the volumes of the cone and the hemisphere are equal, then \( V = \frac{2}{3} \pi R^{3} \); on the other hand, \( V = \frac{1}{3} \pi R^{2} h \), where \( h \) is the height of the cone, i.e., \( h = 2 R \). We have \( S_{\text {side }} = \pi R l \), where \( l = \sqrt{R^{2} + h^{2}} = R \sqrt{5} \). Therefore, \( S_{\text {side }} = \pi R^{2} \sqrt{5} \).

Answer: \( \pi R^{2} \sqrt{5} \).

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-467.jpg?height=483&width=381&top_left_y=499&top_left_x=54)

Fig. 2.20