The total surface area of a cube is 486 square centimetres. What is the sum of the lengths of all its edges (in centimetres)?

Introduction / Context:
This question connects the total surface area of a cube to the sum of its edge lengths. It tests whether you can work backwards from surface area to side length and then use the fact that a cube has 12 equal edges to find the total length of all edges.

Given Data / Assumptions:

• Total surface area (TSA) of the cube = 486 square centimetres.
• A cube has 6 faces, all squares of side length a.
• A cube has 12 edges, each of length a.
• We need the sum of the lengths of all edges.

Concept / Approach:
The total surface area of a cube with side length a is TSA = 6a^2. From this, we can solve for a. Once we know a, the sum of the lengths of all edges is 12a. The problem reduces to solving a simple quadratic equation followed by a multiplication.

Step-by-Step Solution:
Step 1: Let a be the length of one edge of the cube. Step 2: Use the TSA formula: 6a^2 = 486. Step 3: Divide both sides by 6 to solve for a^2: a^2 = 486 / 6 = 81. Step 4: Take the positive square root (since length is positive): a = √81 = 9 cm. Step 5: A cube has 12 edges, each of length a, so the sum of all edge lengths = 12a. Step 6: Substitute a = 9 cm: 12 * 9 = 108 cm.

Verification / Alternative check:
We can check by recomputing the TSA from the found edge length. If a = 9 cm, then TSA = 6a^2 = 6 * 81 = 486 square centimetres, which matches the given value. This confirms that a is correct and that the computed sum of edge lengths, 108 cm, is consistent with the given data.

Why Other Options Are Wrong:
54 cm would correspond to a side length of 4.5 cm (since 12 * 4.5 = 54), which would give TSA = 6 * 4.5^2 = 6 * 20.25 = 121.5 square centimetres, not 486. 162 cm and 216 cm correspond to even larger side lengths, which would give total surface areas much greater than 486 square centimetres. Only 108 cm matches the cube whose surface area is 486 square centimetres.

Common Pitfalls:
Some learners confuse surface area with volume and attempt to use a^3 instead of 6a^2. Others might miscount the number of edges, using 8 or 6 instead of 12, which leads to incorrect sums. Remember that a cube has 6 faces, 8 vertices, and 12 edges, and that TSA depends on a^2 while the sum of edges depends linearly on a.

Final Answer:
The sum of the lengths of all edges of the cube is 108 cm.