The volume of a cube is numerically equal to the sum of its edges. What is its total surface area in square units?

### Concept & Formula A cube consists of 12 equal edges. By setting the formula for volume equal to the formula for the sum of all edges, we can solve for the edge length (or its square), which directly leads to the surface area. $$Volume = a^3$$ $$Sum\ of\ edges = 12a$$ $$Total\ Surface\ Area = 6a^2$$ ### Step-by-Step Solution 1. **Set up the given relation:** $Volume = Sum\ of\ edges$ $a^3 = 12a$ 2. **Solve for $a^2$:** Since a physical edge $a$ must be greater than 0, divide both sides by $a$. $a^2 = 12$ (Note: We don't need to find $a = \sqrt{12}$ because the surface area formula only requires $a^2$.) 3. **Calculate Total Surface Area:** $Surface Area = 6a^2$ Substitute $a^2 = 12$ directly: $Surface Area = 6 \times 12 = 72$ square units. ### Exam Strategy & Shortcut Recognize that you do not always need to isolate the variable completely. Finding $a^2 = 12$ is enough since the target expression ($6a^2$) uses $a^2$. Avoiding the square root saves time and reduces calculation errors. ### Common Pitfall Mistakenly thinking the "sum of its edges" refers to just the perimeter of one face ($4a$) rather than the total framework of the cube (12 edges $\rightarrow 12a$). Setting $a^3 = 4a$ leads to an incorrect surface area of 24. ### Final Answer Therefore, the correct answer is **72**.