Find the total surface area of a cube whose volume is 729 cm³.

Step-by-step solution

1. 1

   See why the edge length is the bridge. Volume and surface area are not directly proportional, so you cannot scale one into the other. Both are, however, determined by the single edge length $a$, so recover $a$ first and then rebuild the surface area from it.

2. 2

   Invert the volume formula. For a cube, $V=a^3$, so

   $a=\sqrt[3]{V}=\sqrt[3]{729}$

   Recognise $729$ as a perfect cube: $9^3=9\times 9\times 9=81\times 9=729$. Hence

   $a=9\ \text{cm}$

3. 3

   Apply the surface-area formula. A cube has $6$ congruent square faces, each of area $a^2$:

   $S=6a^2$

4. 4

   Substitute and evaluate.

   $S=6\times 9^2=6\times 81=486\ \text{cm}^2$

   $\boxed{486\ \text{cm}^2}$

5. 5

   Check the units and the magnitude. Volume carries $\text{cm}^3$ and area carries $\text{cm}^2$, so the answer is correctly in $\text{cm}^2$. As a rough check, one face is $81\ \text{cm}^2$ and there are six of them, so the answer must sit between $80$ and $500$ — and $486$ does.