A circular rug has a circumference of 52pi feet. What is the area, in square feet, of the rug?26pi52pi338pi676pi2{,}704pi

1. TRANSLATE the problem information

• Given information:
  • Circumference = \(\mathrm{52\pi}\) feet
  • Need to find area in square feet

2. INFER the solution strategy

• To find area, I need the radius (\(\mathrm{A = \pi r^2}\))
• I can get radius from circumference using \(\mathrm{C = 2\pi r}\)
• Strategy: Circumference → radius → area

3. SIMPLIFY to find the radius

• Start with circumference formula: \(\mathrm{C = 2\pi r}\)
• Substitute known value: \(\mathrm{52\pi = 2\pi r}\)
• Divide both sides by \(\mathrm{2\pi}\): \(\mathrm{r = 52\pi \div 2\pi = 26}\) feet

4. SIMPLIFY to find the area

• Use area formula: \(\mathrm{A = \pi r^2}\)
• Substitute \(\mathrm{r = 26}\): \(\mathrm{A = \pi(26)^2}\)
• Calculate: \(\mathrm{A = \pi(676) = 676\pi}\) square feet

Answer: D (\(\mathrm{676\pi}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between circumference and area through radius. Instead, they might think the circumference value can be directly used as the area or get confused about which formula to use when.

This may lead them to select Choice B (\(\mathrm{52\pi}\)) - simply using the given circumference as the area.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the radius correctly (\(\mathrm{r = 26}\)) but forget to square it in the area formula, calculating \(\mathrm{A = \pi r}\) instead of \(\mathrm{A = \pi r^2}\).

This may lead them to select Choice A (\(\mathrm{26\pi}\)).

Third Most Common Error:

Conceptual confusion: Students mistakenly use the circumference value (52) as the radius in the area formula, calculating \(\mathrm{A = \pi(52)^2 = 2,704\pi}\).

This may lead them to select Choice E (\(\mathrm{2,704\pi}\)).

The Bottom Line:

This problem tests whether students can connect two circle formulas through their common element (radius) rather than just memorizing isolated formulas. The key insight is recognizing that circumference gives you radius, and radius gives you area.