If the volume of a cube is 729 cm$^3$, then the surface area of the cube will be

### Concept & Strategy We are given the volume of a cube and need to find its surface area. The bridge between these two properties is the side length ($a$). First, find $a$ from the volume, then use it to calculate the surface area. $$Volume = a^3$$ $$Surface Area = 6a^2$$ ### Step-by-Step Solution 1. **Find Side Length ($a$):** $a^3 = 729$ Taking the cube root of both sides: $a = \sqrt[3]{729}$ $a = 9$ cm. 2. **Calculate Surface Area:** $Surface Area = 6a^2$ $Surface Area = 6 \times (9)^2$ $Surface Area = 6 \times 81$ $Surface Area = 486$ cm$^2$. ### Exam Strategy & Shortcut Unit digit strategy: Volume is $729$, ending in 9. The only single-digit number whose cube ends in 9 is 9 ($9^3 = 729$). So, side $a = 9$. Surface area ends with the unit digit of $6 \times 9^2 \rightarrow 6 \times 1 = 6$. Looking at the options, three end in 6, so we must calculate. $6 \times 80 = 480$, $6 \times 1 = 6 \rightarrow 486$. ### Common Pitfall Forgetting to multiply by 6 for the surface area, or confusing square roots and cube roots (e.g., trying to find $\sqrt{729}$ instead of $\sqrt[3]{729}$). ### Final Answer Therefore, the correct answer is **486 cm$^2$**.