Regular hexagon A has an area of 36 square centimeters. A second regular hexagon, B, is similar to hexagon A....

1. TRANSLATE the problem information

• Given information:
  • Regular hexagon A: \(\mathrm{area = 36\text{ cm}^2}\)
  • Regular hexagon B: similar to A
  • \(\mathrm{Perimeter(B) = Perimeter(A)}\)
• What this tells us: We need to find how the equal perimeters affect the areas

2. TRANSLATE the perimeter condition

• For any regular hexagon with side length s: \(\mathrm{Perimeter = 6s}\)
• Let \(\mathrm{s_A}\) = side length of hexagon A, \(\mathrm{s_B}\) = side length of hexagon B
• Equal perimeters means: \(\mathrm{6s_A = 6s_B}\)

3. SIMPLIFY the perimeter equation

• \(\mathrm{6s_A = 6s_B}\)
• Divide both sides by 6: \(\mathrm{s_A = s_B}\)
• The hexagons have identical side lengths!

4. INFER the similarity relationship

• For similar figures, we compare using the similarity ratio \(\mathrm{k = \frac{s_B}{s_A}}\)
• Since \(\mathrm{s_A = s_B}\), we have \(\mathrm{k = \frac{s_B}{s_A} = 1}\)
• When \(\mathrm{k = 1}\), the figures are not just similar—they're congruent (identical)

5. INFER the area relationship

• For similar figures: \(\mathrm{Area\text{ }ratio = k^2}\)
• Here: \(\mathrm{Area\text{ }ratio = 1^2 = 1}\)
• This means: \(\mathrm{\frac{Area(B)}{Area(A)} = 1}\), so \(\mathrm{Area(B) = Area(A)}\)

6. APPLY CONSTRAINTS to find the final answer

• Since \(\mathrm{Area(A) = 36\text{ cm}^2}\), we get \(\mathrm{Area(B) = 36\text{ cm}^2}\)

Answer: B) 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly find that the side lengths are equal but then assume that similar figures must have different areas. They may think "if B is similar to A, it must be larger or smaller" and look for an area that's not 36.

This misconception leads them to select Choice C (72) or Choice D (144), thinking the area must be different even though the figures are identical.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students may misinterpret "similar" to mean "different in size" and assume the hexagons cannot be the same size. They skip the crucial step of setting up the perimeter equation and instead guess that one hexagon must be larger.

This leads to confusion and random selection among the larger answer choices.

The Bottom Line:

The key insight is that "similar" includes the special case where figures are congruent (\(\mathrm{k = 1}\)). When two similar figures have equal perimeters, they must be identical, making this problem simpler than it initially appears.