The sum of the lengths of all 12 edges of a cube is 84 cm. What is the volume of the cube (in cubic centimetres)?

Introduction / Context:
This is a straightforward cube mensuration problem. It gives the total length of all edges of a cube and asks for the volume. The question checks whether you know how many edges a cube has, how to relate edge length to volume, and how to work backwards from a total edge length to the side length.

Given Data / Assumptions:

• A cube has 12 edges, all of equal length.
• The sum of the lengths of all edges is 84 cm.
• We need to find the volume of the cube in cubic centimetres.

Concept / Approach:
If each edge of the cube has length a, and there are 12 edges in total, then the sum of edge lengths is 12a. Setting 12a equal to the given total 84 cm, we solve for a. Once a is known, the volume V of the cube is given by V = a^3. This problem is mainly about setting up a simple linear equation and then computing a cube of an integer.

Step-by-Step Solution:
Step 1: Let a be the length of each edge of the cube in centimetres. Step 2: There are 12 edges, so total length of edges = 12a. Step 3: We are told 12a = 84. Step 4: Solve for a: a = 84 / 12 = 7 cm. Step 5: Volume of a cube is V = a^3. Step 6: Compute V = 7^3 = 7 * 7 * 7 = 343 cubic centimetres.

Verification / Alternative check:
To verify, start from the calculated edge length a = 7 cm. The sum of all edges should be 12 * 7 = 84 cm, which matches the given data. The volume 343 cubic centimetres also makes sense because 7 cm is a common side length whose cube is 343, a standard mental math fact (7^3 = 343).

Why Other Options Are Wrong:
686 cubic centimetres is 2 * 343 and might arise from incorrectly doubling the cube of the side length. 171.5 cubic centimetres and 514.5 cubic centimetres involve decimals that do not correspond to a simple integer cube and would not result from a neat edge length when the total edge length is 84 cm. Only 343 cubic centimetres is exactly a^3 for a = 7, consistent with the given information.

Common Pitfalls:
Some learners misremember the number of edges in a cube and might use 8 (number of vertices) or 6 (number of faces) instead of 12. This leads to wrong calculations for the side length. Others might mistakenly compute volume as 6a^3 or a^2. Remember that volume measures space inside the cube and is always given by a^3, while 6a^2 corresponds to surface area, not volume.

Final Answer:
The volume of the cube is 343 cubic centimetres.