2.52. The radius of the base of the cone is $R$, and the angle of the sector of its lateral surface is $90^{\circ}$. Determine the volume of the cone.
2.52. Let $l$ be the slant height of the cone. Since the length of the arc of the lateral surface development is equal to the circumference of the base, we have $2 \pi R=\frac{2 \pi l}{4}$ (by the condition, the development is a quarter circle), from which $l=4 R$. Let's find the height of the cone: $h=\sqrt{l^{2}-R^{2}}=$ $=\sqrt{16 R^{2}-R^{2}}=R \sqrt{15} \cdot$ Finally, we get

$$
V=\frac{1}{3} \pi R^{2} \cdot R \sqrt{15}=\frac{\pi R^{3} \sqrt{15}}{3}
$$

Answer: $\frac{\pi R^{3} \sqrt{15}}{3}$.