The volume of a cube is 729 cubic centimetres. What is the total surface area of the cube in square centimetres?

Introduction / Context:
This problem links the volume of a cube to its surface area. Once the edge length of a cube is known, its volume and surface area can both be calculated easily. In this question, the volume is given and we must reverse the volume formula to find the edge length first, then apply the surface area formula. Such questions test your familiarity with powers and roots, particularly cube roots, and formula application.

Given Data / Assumptions:

• Volume of the cube, V = 729 cm^3.
• Edge length (side) of the cube = a cm.
• We must find the total surface area in square centimetres.

Concept / Approach:
For a cube of side a, the volume is:
V = a^3 The total surface area (TSA) is:
TSA = 6 * a^2 First, we find a by taking the cube root of the given volume. Once a is known, we substitute it into the surface area formula to obtain the result.

Step-by-Step Solution:
Given V = 729 cm^3. For a cube, V = a^3, so a^3 = 729. We recognize 729 = 9^3 (since 9 * 9 * 9 = 729). Therefore, a = 9 cm. Total surface area TSA = 6 * a^2. Compute a^2 = 9^2 = 81. TSA = 6 * 81 = 486 square centimetres.

Verification / Alternative check:
We can reverse the calculation: if the cube has side 9 cm, its volume would be 9^3 = 729 cm^3, which matches the given volume. This confirms that a = 9 cm is correct. Then TSA = 6 * 81 = 486 cm^2 fits perfectly with the formula and the computed side length, so our answer is consistent in both directions.

Why Other Options Are Wrong:
The values 456, 446, 476 and 500 sq.cm all correspond to incorrectly squaring the side or using an approximate side length that is not the true cube root of 729. For example, 500 sq.cm would require a non-integer side length. Only 486 sq.cm comes from the correct side length of 9 cm and the standard surface area formula for a cube.

Common Pitfalls:
Some students mistakenly take the square root instead of the cube root of the volume, leading to a side length of sqrt(729) rather than cube root. Others confuse the formulas for volume and surface area or forget that a cube has 6 identical faces. Careful recall of definitions and formulas helps avoid these errors.

Final Answer:
The total surface area of the cube is 486 sq.cm.