Two similar regular hexagons, H and K, have perimeters 30n and 210n, respectively, where n is a positive constant. The...

1. TRANSLATE the problem information

• Given information:
  • Hexagon H has perimeter 30n
  • Hexagon K has perimeter 210n
  • The hexagons are similar
  • Need to find: Area of K ÷ Area of H

2. INFER the linear scale factor

• Since the hexagons are similar, their perimeters are proportional to their corresponding side lengths
• Linear scale factor = Perimeter of K ÷ Perimeter of H = \(210\mathrm{n} \div 30\mathrm{n} = 7\)
• This means each side of hexagon K is 7 times longer than the corresponding side of hexagon H

3. INFER the area relationship

• For similar figures, areas scale by the square of the linear scale factor
• If the linear scale factor is 7, then the area scale factor is \(7^2 = 49\)
• Therefore, the area of hexagon K is 49 times the area of hexagon H

Answer: (C) 49

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize the perimeter ratio is 7 but then use 7 as the final answer instead of squaring it for the area ratio.

They correctly find that \(210\mathrm{n} \div 30\mathrm{n} = 7\), but then incorrectly conclude that the area of K is 7 times the area of H. They forget that areas scale by \(\mathrm{k}^2\), not \(\mathrm{k}\).

This leads them to select Choice (A) (7).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or understand the area scaling rule for similar figures.

They might try to work with actual hexagon area formulas or get confused about how similarity affects area versus perimeter. Without the key insight that areas scale by \(\mathrm{k}^2\), they cannot systematically solve the problem.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students truly understand the difference between linear scaling (which affects lengths and perimeters) and area scaling (which goes by the square of the linear factor). The trap is that the linear scale factor (7) appears as answer choice (A), catching students who don't complete the area scaling step.