A right square pyramid has a volume of 3,200 cubic meters and a height of 24 meters. What is the...

1. TRANSLATE the problem information

• Given information:
  • Volume = 3,200 cubic meters
  • Height = 24 meters
  • Right square pyramid
• Need to find: Area of one triangular face

2. INFER the strategic approach

• To find triangular face area, we need base length and slant height
• We can get base length from the volume, then use geometry to find slant height
• Work backwards: volume → base area → side length → slant height → face area

3. SIMPLIFY to find the base area

Using \(\mathrm{V = \frac{1}{3} \times base\ area \times height}\):

• \(\mathrm{3{,}200 = \frac{1}{3} \times B \times 24}\)
• \(\mathrm{3{,}200 = 8B}\)
• \(\mathrm{B = 400}\) square meters

4. INFER the side length from base area

Since the base is square:

• \(\mathrm{s^2 = 400}\)
• \(\mathrm{s = 20}\) meters

5. VISUALIZE the slant height geometry

The slant height connects the apex to the midpoint of a base edge, forming a right triangle with:

• Vertical leg = height = 24 meters
• Horizontal leg = distance from center to edge midpoint = \(\mathrm{s/2 = 10}\) meters

6. SIMPLIFY using Pythagorean theorem

• \(\mathrm{l^2 = 24^2 + 10^2 = 576 + 100 = 676}\)
• \(\mathrm{l = 26}\) meters (use calculator if needed)

7. Calculate triangular face area

• \(\mathrm{Area = \frac{1}{2} \times base \times slant\ height = \frac{1}{2} \times 20 \times 26 = 260}\) square meters

Answer: B. 260

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to find the triangular face area directly without recognizing they need the slant height first, or they confuse the pyramid's height with the slant height.

They might incorrectly use the pyramid height (24) as the triangle's height: \(\mathrm{\frac{1}{2} \times 20 \times 24 = 240}\), leading them to select Choice A (240).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "triangular face" means and calculate something other than the slant triangle area.

This leads to confusion and random guessing among the remaining choices.

The Bottom Line:

This problem requires understanding that finding surface areas of 3D shapes often involves finding intermediate measurements first. The key insight is recognizing that slant height ≠ pyramid height and requires the Pythagorean theorem.