The volume of a sphere of radius $r$ is obtained by multiplying its surface area by

### Concept & Formula The relationship between the volume and surface area of a sphere involves comparing their respective formulas. Volume of a sphere: $$V = \frac{4}{3}\pi r^3$$ Surface Area of a sphere: $$S = 4\pi r^2$$ ### Step-by-Step Solution 1. Let the required multiplier be $x$. 2. According to the problem statement: Volume = Surface Area $\times x$ 3. Substitute the standard formulas into the equation: $$\frac{4}{3}\pi r^3 = (4\pi r^2) \times x$$ 4. Isolate $x$ by dividing both sides by $4\pi r^2$: $$x = \frac{\frac{4}{3}\pi r^3}{4\pi r^2}$$ 5. Simplify the expression by canceling out common terms ($4$, $\pi$, and $r^2$): $$x = \frac{r}{3}$$ ### Exam Strategy & Shortcut Memorizing the ratio $\frac{V}{S} = \frac{r}{3}$ for a sphere is a massive time-saver. By knowing this core property, you can immediately conclude that multiplying $S$ by $\frac{r}{3}$ yields $V$ without any calculation. ### Common Pitfall Students frequently select $\frac{4}{3}$ by mistakenly identifying it as the unique coefficient of the volume formula, ignoring the fact that it must multiply the full surface area expression ($4\pi r^2$), which requires further division. ### Final Answer Therefore, the correct answer is **$\frac{r}{3}$**.