Find the total surface area of a cube whose volume is 729 cubic centimetres. Choose from 512, 486, 294 or 384 square centimetres.

Step-by-step solution

1. 1

   Go through the edge length — never directly from volume to area. Volume and surface area are related only through the edge $a$, by

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

   There is no shortcut that skips $a$, because the two formulas have different powers.

2. 2

   Take the cube root of the volume.

   $a=\sqrt[3]{729}=9\ \text{cm},$

   since $9^3=9\cdot 9\cdot 9=81\cdot 9=729$. Recognising $729=3^6=(3^2)^3$ gets you there instantly.

3. 3

   Find the area of one face. Each face is a square of side $a$:

   $a^{2}=9^{2}=81\ \text{cm}^2.$

4. 4

   Multiply by the six faces.

   $S=6\cdot 81=486\ \text{cm}^2,$

   which is option 2.

5. 5

   Check the distractors. $512=8^3$ is a volume, not an area, and belongs to a cube of edge $8$; $384=6\cdot 8^2$ is the surface area of that wrong cube; $294=6\cdot 7^2$ belongs to edge $7$. Each corresponds to mis-taking the cube root, which is exactly the step this question is testing.