Question:The perimeter of a regular hexagon is 456 centimeters. The apothem of this hexagon is ksqrt(3) centimeters, where k is...

1. TRANSLATE the problem information

• Given information:
  • Regular hexagon perimeter = 456 cm
  • Apothem = \(\mathrm{k\sqrt{3}\ cm}\) (where k is unknown)
  • Need to find the value of k

• What this tells us: We have a hexagon with 6 equal sides, and we know the total distance around it.

2. INFER the approach

• To find k, I need to determine what the apothem actually equals in numerical form
• This means I need the side length first, then I can use the apothem formula
• Strategy: Find side length → Calculate apothem → Set equal to \(\mathrm{k\sqrt{3}}\) → Solve for k

3. TRANSLATE to find the side length

• Since a regular hexagon has 6 equal sides:
• Side length = Total perimeter ÷ 6 = 456 ÷ 6 = 76 cm

4. INFER which formula to use

• For a regular hexagon, the apothem (distance from center to middle of any side) has a specific relationship to side length
• The formula is: apothem = \(\mathrm{\frac{s\sqrt{3}}{2}}\), where s is the side length

5. SIMPLIFY the apothem calculation

• Apothem = \(\mathrm{\frac{76\sqrt{3}}{2} = 38\sqrt{3}\ cm}\)

6. SIMPLIFY to solve for k

• We know the apothem equals both \(\mathrm{38\sqrt{3}}\) and \(\mathrm{k\sqrt{3}}\)
• Setting them equal: \(\mathrm{k\sqrt{3} = 38\sqrt{3}}\)
• Dividing both sides by \(\mathrm{\sqrt{3}}\): \(\mathrm{k = 38}\)

Answer: 38

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "regular hexagon with perimeter 456" means each side is \(\mathrm{456÷6 = 76\ cm}\). Some students might try to use 456 directly in the apothem formula, or get confused about what "regular" means in this context.

This leads to incorrect calculations and confusion, causing them to abandon systematic solution and guess.

Second Most Common Error:

Missing conceptual knowledge: Students may not remember or know the apothem formula for a regular hexagon: apothem = \(\mathrm{\frac{s\sqrt{3}}{2}}\). Without this formula, they cannot connect the side length to the apothem value.

This causes them to get stuck after finding the side length and resort to guessing.

The Bottom Line:

This problem requires both solid conceptual knowledge (hexagon properties and apothem formula) and strong translation skills to convert the perimeter into usable side length information. Students who struggle with either component will find it difficult to make progress systematically.