If the ratio of the volume of two spheres is 1 : 8, find the ratio of their surface areas.

Given,

Ratio of the volumes of the two spheres is 1 : 8

$\therefore \dfrac{\text{Volume of sphere 1}}{\text{Volume of sphere 2}} = \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{\dfrac{4}{3} π\text{R}^3}{\dfrac{4}{3} π\text{r}^3} = \dfrac{1}{8} \\[1em] \Rightarrow \dfrac{\text{R}^3}{\text{r}^3} = \dfrac{1^3}{2^3} \\[1em] \Rightarrow \dfrac{\text{R}}{\text{r}} = \dfrac{1}{2}$

Surface area of sphere = 4πr²

$\therefore \dfrac{\text{Surface area of sphere 1}}{\text{Surface area of sphere 2}} = \dfrac{4π\text{R}^2}{4π\text{r}^2} \\[1em] = \Big(\dfrac{\text{R}}{\text{r}}\Big)^2 \\[1em] = \Big(\dfrac{1}{2}\Big)^2 \\[1em] = \dfrac{1}{4}.$

Hence, the ratio of the surface areas of two spheres is 1 : 4.