Express Volume Ratio

The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$. The ratio of the volumes of two spheres is the cube of the ratio of their radii.

$\frac{V_1}{V_2} = \frac{r_1^3}{r_2^3} = \frac{27}{64}$

Find Radius Ratio

Taking the cube root of the volume ratio gives the ratio of their radii.

$\frac{r_1}{r_2} = \sqrt[3]{\frac{27}{64}} = \frac{3}{4}$

Express Surface Area Ratio

The surface area of a sphere is given by $S = 4\pi r^2$. The ratio of their surface areas is the square of the ratio of their radii.

$\frac{S_1}{S_2} = \frac{r_1^2}{r_2^2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16}$