If the total length of diagonals of a cube is 12 cm, then what is the total length of the edges of the cube?

### Concept & Logic A cube has 4 main body diagonals that pass through its center, connecting opposite vertices. The "total length of diagonals" refers to the sum of these 4 lengths. A cube also has 12 equal edges. $$Total\ Diagonals = 4 \times (a\sqrt{3})$$ $$Total\ Edges = 12 \times a$$ ### Step-by-Step Solution 1. **Find the length of one diagonal:** Total length of 4 diagonals $= 12$ cm. Length of one diagonal $= \frac{12}{4} = 3$ cm. 2. **Find the edge length ($a$):** $a\sqrt{3} = 3$ $a = \frac{3}{\sqrt{3}}$ Rationalizing the denominator: $a = \frac{3\sqrt{3}}{3} = \sqrt{3}$ cm. 3. **Calculate total length of edges:** Total edges $= 12 \times a$ Total edges $= 12 \times \sqrt{3} = 12\sqrt{3}$ cm. ### Exam Strategy & Shortcut Notice the relationship algebraically: Total diagonals $= 4a\sqrt{3}$. We want $12a$. To get $12a$ from $4a\sqrt{3}$, multiply by $\sqrt{3}$: $(4a\sqrt{3}) \times \sqrt{3} = 4a \times 3 = 12a$. Since the total diagonals sum is 12, just multiply it by $\sqrt{3} \rightarrow 12\sqrt{3}$. ### Common Pitfall Assuming "diagonals" refers to the face diagonals (there are 12 of them, each $a\sqrt{2}$) instead of the 4 main body diagonals. The phrasing "diagonals of a cube" conventionally implies the body diagonals. ### Final Answer Therefore, the correct answer is **$12\sqrt{3}$ cm**.