The total length of all edges of a cube is 6b, where b is a positive constant. Which of the...

1. TRANSLATE the problem information

• Given information:
  • Total length of all edges = 6b
  • Need to find: area of one face

2. INFER the approach

• To find face area, I need the length of one edge first
• Each face of a cube is a square, so face area = (edge length)²
• A cube has 12 equal edges

3. TRANSLATE the edge relationship

• If all edges have the same length s, then:

\(\mathrm{12s = 6b}\)

4. SIMPLIFY to find edge length

• Divide both sides by 12:

\(\mathrm{s = \frac{6b}{12} = \frac{b}{2}}\)

5. SIMPLIFY to find face area

• Area of square face = \(\mathrm{s^2}\)
• Area = \(\mathrm{(\frac{b}{2})^2 = \frac{b^2}{4}}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about cube properties: Students might forget that a cube has 12 edges, thinking it has 6 edges (confusing edges with faces) or 8 edges (confusing edges with vertices).

If they think a cube has 6 edges, they would calculate: edge length = \(\mathrm{\frac{6b}{6} = b}\), leading to face area = \(\mathrm{b^2}\). If they think it has 8 edges, they would get: edge length = \(\mathrm{\frac{6b}{8} = \frac{3b}{4}}\), leading to face area = \(\mathrm{\frac{9b^2}{16}}\). Neither of these matches any answer choice, so this leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{(\frac{b}{2})^2}\) but make algebraic errors, such as getting \(\mathrm{\frac{b^2}{2}}\) instead of \(\mathrm{\frac{b^2}{4}}\) by incorrectly applying the exponent to the fraction.

This may lead them to select Choice (D) (\(\mathrm{\frac{b^2}{2}}\)).

The Bottom Line:

This problem tests whether students truly understand three-dimensional geometry properties and can systematically work through multi-step algebraic relationships without rushing.