The total surface area of a cube is 864 square inches. What is the length, in inches, of an edge...

1. TRANSLATE the problem information

• Given information:
  • Total surface area of cube = 864 square inches
  • Need to find: edge length

• What this tells us: We need to use the relationship between surface area and edge length

2. INFER the approach

• A cube has 6 identical square faces
• If each edge is length s, then each face has area \(\mathrm{s^2}\)
• Strategy: Use surface area formula \(\mathrm{SA = 6s^2}\) and solve for s

3. TRANSLATE into mathematical equation

• Set up: \(\mathrm{864 = 6s^2}\)

4. SIMPLIFY to solve for the edge length

• Divide both sides by 6: \(\mathrm{s^2 = 864 ÷ 6 = 144}\)
• Take the square root: \(\mathrm{s = \sqrt{144} = 12}\)

Answer: 12 inches (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not knowing or misremembering the cube surface area formula

Students might think a cube only has 4 faces (confusing it with a square) or forget that the formula is \(\mathrm{SA = 6s^2}\), not \(\mathrm{SA = s^2}\). This leads to setting up wrong equations like:

• \(\mathrm{864 = 4s^2}\) (thinking cube has 4 faces) → \(\mathrm{s^2 = 216}\) → \(\mathrm{s = \sqrt{216} ≈ 14.7}\)
• \(\mathrm{864 = s^2}\) (forgetting the factor of 6) → \(\mathrm{s = \sqrt{864} ≈ 29.4}\)

This confusion causes them to get stuck and guess, or select Choice D (14) if they approximate \(\mathrm{\sqrt{216}}\).

Second Most Common Error:

Weak SIMPLIFY execution: Making arithmetic errors in the division or square root steps

Students might:

• Calculate \(\mathrm{864 ÷ 6}\) incorrectly (getting 154 instead of 144)
• Not recognize that \(\mathrm{\sqrt{144} = 12}\), leading to calculator errors or wrong approximations

This may lead them to select Choice B (11) or other incorrect choices based on their calculation mistakes.

The Bottom Line:

Success requires both knowing the specific cube surface area formula (\(\mathrm{SA = 6s^2}\)) and executing clean arithmetic. The most challenging part is often remembering that a cube has 6 faces, not 4.