A right square pyramid has a volume of 1,728 cubic centimeters and the area of its base is 576 square...

1. TRANSLATE the problem information

• Given information:
  • Right square pyramid with volume 1,728 cm³
  • Base area is 576 cm²
  • Need to find slant height of a triangular face

• What this tells us: We need to work backwards from volume to find dimensions, then use geometry to find slant height

2. INFER the solution strategy

• Key insight: Slant height requires knowing the pyramid's height and base dimensions
• Strategy: Find base side length → Find pyramid height → Use right triangle relationship for slant height

3. SIMPLIFY to find the base side length

• Since base area = side², we have:
  \(576 = \mathrm{side}^2\)
  \(\mathrm{side} = \sqrt{576} = 24\text{ cm}\)

4. SIMPLIFY to find the pyramid height

• Using volume formula V = (1/3) × base area × height:
  \(1,728 = \frac{1}{3} \times 576 \times \mathrm{height}\)
  \(1,728 = 192 \times \mathrm{height}\)
  \(\mathrm{height} = 9\text{ cm}\)

5. VISUALIZE the slant height geometry

• The slant height creates a right triangle with:
  • Vertical leg: pyramid height = 9 cm
  • Horizontal leg: distance from center of base to midpoint of edge = \(\frac{24}{2} = 12\text{ cm}\)
  • Hypotenuse: slant height (what we want)

6. SIMPLIFY using Pythagorean theorem

• \(\mathrm{slant\ height} = \sqrt{9^2 + 12^2}\)
  \(= \sqrt{81 + 144}\)
  \(= \sqrt{225}\)
  \(= 15\text{ cm}\)

Answer: C (15)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "slant height of a triangular face" means and attempt to find the pyramid's height instead, or confuse slant height with the edge length of the pyramid.

This leads them to select Choice A (9) - the pyramid height rather than the slant height.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need multiple steps but attempt to use the wrong geometric relationship, such as trying to find slant height directly from volume without first determining the pyramid height and base dimensions.

This leads to confusion and incorrect calculations, potentially causing them to select Choice B (12) - the distance from center to edge.

The Bottom Line:

This problem requires students to understand that finding slant height is a multi-step geometric process that builds on the pyramid's fundamental measurements, not a direct calculation from the given information.