A cube has a surface area of 54 square meters. What is the volume, in cubic meters, of the cube?

1. TRANSLATE the problem information

• Given information:
  • Surface area of cube = 54 square meters
  • Need to find volume in cubic meters

2. INFER the solution strategy

• Key insight: You can't find volume directly from surface area
• Strategy: Use surface area to find side length first, then calculate volume
• This works because both formulas depend on the same side length

3. SIMPLIFY to find the side length

• Surface area formula: \(\mathrm{SA = 6s^2}\) (6 faces, each with area \(\mathrm{s^2}\))
• Set up equation: \(\mathrm{6s^2 = 54}\)
• Divide both sides by 6: \(\mathrm{s^2 = 9}\)
• Take square root: \(\mathrm{s = 3}\) meters

4. SIMPLIFY to find the volume

• Volume formula: \(\mathrm{V = s^3}\)
• Substitute: \(\mathrm{V = 3^3 = 27}\) cubic meters

Answer: B. 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to find a direct relationship between surface area and volume without recognizing they need to find side length as an intermediate step.

They might attempt to divide 54 by something random or look for a pattern in the answer choices, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when solving \(\mathrm{6s^2 = 54}\), perhaps getting \(\mathrm{s^2 = 6}\) instead of \(\mathrm{s^2 = 9}\), which leads to \(\mathrm{s = \sqrt{6} \approx 2.45}\), and then \(\mathrm{V \approx 14.7}\).

Since this doesn't match any answer choice exactly, this causes them to get stuck and guess between the closest options.

The Bottom Line:

This problem requires students to see the connection between two different formulas through a shared variable (side length). Success depends on strategic thinking about the solution pathway, not just knowing the formulas.