The diameter of the Moon is approximately one-fourth of the diameter of the Earth. We need to find the ratio of their surface areas.
The surface area \( A \) of a sphere is given by the formula: \[ A = 4 \pi r^2 \] where \( r \) is the radius of the sphere.
Let the diameter of the Earth be \( D_{\text{Earth}} \) and the diameter of the Moon be \( D_{\text{Moon}} \). According to the problem: \[ D_{\text{Moon}} = \frac{1}{4} D_{\text{Earth}} \] The radius is half of the diameter, so: \[ r_{\text{Moon}} = \frac{1}{4} r_{\text{Earth}} \]
- Surface area of the Earth: \[ A_{\text{Earth}} = 4 \pi r_{\text{Earth}}^2 \] - Surface area of the Moon: \[ A_{\text{Moon}} = 4 \pi r_{\text{Moon}}^2 = 4 \pi \left(\frac{1}{4} r_{\text{Earth}}\right)^2 = 4 \pi \times \frac{1}{16} r_{\text{Earth}}^2 \] \[ A_{\text{Moon}} = \frac{1}{4} A_{\text{Earth}} \]
The ratio of the surface area of the Moon to the Earth is: \[ \frac{A_{\text{Moon}}}{A_{\text{Earth}}} = \frac{\frac{1}{4} A_{\text{Earth}}}{A_{\text{Earth}}} = \frac{1}{4} \]
The ratio of the surface area of the Moon to the Earth is: \[ \boxed{\frac{1}{16}} \]