Let the radius of the base of both the hemisphere and the cone be \(r\) units. Also, let the height of both be equal to the radius of the hemisphere, i.e., \(r\) units. ∴ The slant height of the cone = \(\sqrt{r^2 + r^2} = \sqrt{2}r\) units Now, the ratio of the curved surface area of the hemisphere to that of the cone is: \(= 2\pi r^2 : \pi r \times \sqrt{2}r\) \(= 2 : \sqrt{2} = \sqrt{2} : 1\) ∴ The ratio of the curved surface areas of the hemisphere and the cone is \(\sqrt{2} : 1\)