Find the volume of a cube whose total surface area is 486 square centimetres.

Step-by-step solution

1. 1

   Route the calculation through the edge length. The two cube formulas are

   $S=6a^{2},\qquad V=a^{3},$

   and they share only the edge $a$. Recover $a$ first; there is no direct surface-to-volume formula.

2. 2

   Divide by the number of faces. A cube has six congruent square faces, so one face has area

   $a^{2}=\frac{486}{6}=81\ \text{cm}^2.$

3. 3

   Take the positive square root for the edge.

   $a=\sqrt{81}=9\ \text{cm}.$

   Only the positive root is meaningful for a length.

4. 4

   Cube the edge.

   $V=a^{3}=9^{3}=729\ \text{cm}^3.$

5. 5

   Check by going backwards. With $a=9$, the surface area is $6(81)=486\ \text{cm}^2$ ✓, so the edge is right and the volume follows. Note the tidy pair: $486\ \text{cm}^2$ of surface corresponds to $729\ \text{cm}^3$ of volume — the exact reverse of the companion problem that starts from the volume.