Understanding the Moon-Earth Diameter Ratio

The problem states that the diameter of the Moon is approximately one-fourth the diameter of the Earth. We need to find the ratio of their surface areas based on this information.

Relating Diameter, Radius, and Surface Area

To find the ratio of the surface areas, we first need to understand how diameter, radius, and surface area are related.

• Diameter and Radius: The radius ($R$) of a sphere is half its diameter ($D$). So, $R = \frac{D}{2}$.
• Surface Area Formula: The surface area ($A$) of a sphere is calculated using the formula $A = 4 \pi R^2$.

Step-by-Step Calculation

Let's denote the diameter and radius of the Moon as $D_M$ and $R_M$, and those of the Earth as $D_E$ and $R_E$.

1. Diameter Relationship: We are given that $D_M = \frac{1}{4} D_E$.
2. Radius Calculation:
   • The radius of the Moon is $R_M = \frac{D_M}{2}$. Substituting the given diameter relationship, we get $R_M = \frac{(\frac{1}{4} D_E)}{2} = \frac{1}{4} \left( \frac{D_E}{2} \right)$.
   • Since $R_E = \frac{D_E}{2}$, we can write the Moon's radius in terms of the Earth's radius: $R_M = \frac{1}{4} R_E$. This means the Moon's radius is also one-fourth the Earth's radius.
3. Surface Area Ratio:
   • The surface area of the Moon is $A_M = 4 \pi R_M^2$.
   • The surface area of the Earth is $A_E = 4 \pi R_E^2$.
   • The ratio of their surface areas is $\frac{A_M}{A_E}$.
4. Substituting Radii: We can substitute the relationship $R_M = \frac{1}{4} R_E$ into the ratio formula: $ \frac{A_M}{A_E} = \frac{4 \pi R_M^2}{4 \pi R_E^2} = \frac{R_M^2}{R_E^2} = \left( \frac{R_M}{R_E} \right)^2 $ $ \frac{A_M}{A_E} = \left( \frac{\frac{1}{4} R_E}{R_E} \right)^2 = \left( \frac{1}{4} \right)^2 $
5. Final Ratio: Calculating the square: $ \left( \frac{1}{4} \right)^2 = \frac{1^2}{4^2} = \frac{1}{16} $

Conclusion on Surface Area Ratio

The ratio of the surface area of the Moon to the surface area of the Earth is 1 : 16.