If the radii of two spheres are in the ratio $1 : 4$, then their surface areas are in the ratio

### Concept & Formula The surface area of a sphere is proportional to the square of its radius ($S \propto r^2$). Therefore, the ratio of the surface areas of two spheres is equal to the square of the ratio of their radii. $$\frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2$$ ### Step-by-Step Solution 1. Given the ratio of the radii: $\frac{r_1}{r_2} = \frac{1}{4}$. 2. The formula for the surface area of a sphere is $S = 4\pi r^2$. 3. Write the ratio of their surface areas: $$\frac{S_1}{S_2} = \frac{4\pi r_1^2}{4\pi r_2^2}$$ 4. Cancel out the constant term ($4\pi$): $$\frac{S_1}{S_2} = \frac{r_1^2}{r_2^2} = \left(\frac{r_1}{r_2}\right)^2$$ 5. Substitute the given ratio into the equation: $$\frac{S_1}{S_2} = \left(\frac{1}{4}\right)^2 = \frac{1}{16}$$ 6. The resulting ratio is $1 : 16$. ### Exam Strategy & Shortcut For any 2D measurement (area, surface area) of similar figures, the ratio is simply the square of the 1D measurement (radius, length, perimeter) ratio. Square $1:4$ instantly to get $1:16$. ### Common Pitfall A frequent mistake is to assume the surface area ratio is the same as the radius ratio (yielding $1:4$), or mistakenly applying the volume rule (cubing it to get $1:64$). ### Final Answer Therefore, the correct answer is **$1 : 16$**.