A square patio has a perimeter of 220 centimeters. What is the area, in square centimeters, of the patio? 110...

1. TRANSLATE the problem information

• Given information:
  • Square patio with perimeter = 220 cm
  • Need to find: area in square centimeters

• This tells us we have \(\mathrm{P = 220}\) and need to find \(\mathrm{A}\)

2. INFER the solution strategy

• To find area of a square (\(\mathrm{A = s^2}\)), we need the side length
• We can get side length from perimeter using \(\mathrm{P = 4s}\)
• Strategy: Use perimeter → find side length → calculate area

3. SIMPLIFY to find the side length

• Using \(\mathrm{P = 4s}\) with \(\mathrm{P = 220}\):
  • \(\mathrm{220 = 4s}\)
  • \(\mathrm{s = 220 ÷ 4 = 55\text{ cm}}\)

4. SIMPLIFY to calculate the area

• Using \(\mathrm{A = s^2}\) with \(\mathrm{s = 55}\):
  • \(\mathrm{A = 55^2 = 3,025\text{ cm}^2}\) (use calculator)

Answer: C) 3,025

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about formulas: Students confuse perimeter and area relationships, attempting to use the perimeter value directly or incorrectly applying area formulas.

Some students see "220" and "square" and try shortcuts like dividing by 2 or 4 to get area, not recognizing they need to find side length first through the perimeter formula. This may lead them to select Choice A (110) or Choice B (220).

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly find \(\mathrm{s = 55}\) but make calculation errors when computing \(\mathrm{55^2}\).

They might calculate \(\mathrm{55 × 55}\) incorrectly, potentially getting values that lead them to guess among the answer choices rather than systematically computing the correct value of 3,025.

The Bottom Line:

This problem requires students to recognize the two-step relationship: perimeter gives you side length, which then gives you area. Students who try to jump directly from perimeter to area miss this crucial intermediate step.