A circle in the xy-plane is centered at the origin and has a radius of 6. A sector of this...

1. TRANSLATE the problem information

• Given information:
  • Circle has radius \(\mathrm{r = 6}\)
  • Sector has central angle = \(\mathrm{60°}\)
  • Sector area = \(\mathrm{k\pi}\) (where k is unknown)

• What we need to find: The value of k

2. INFER the approach

• Key insight: A sector's area is a fraction of the total circle's area
• The fraction equals (sector angle)/(full circle angle)
• Strategy: Find total area, then find what fraction the \(\mathrm{60°}\) sector represents

3. Calculate the total circle area

• Use the circle area formula: \(\mathrm{A = \pi r^2}\)
• \(\mathrm{A = \pi(6)^2}\)
  \(\mathrm{= 36\pi}\)

4. INFER the sector's fraction of the circle

• A full circle = \(\mathrm{360°}\)
• Our sector = \(\mathrm{60°}\)
• Fraction = \(\mathrm{\frac{60°}{360°}}\)
  \(\mathrm{= \frac{1}{6}}\)

5. SIMPLIFY to find the sector area

• Sector area = (fraction) × (total area)
• Sector area = \(\mathrm{\frac{1}{6} \times 36\pi}\)
  \(\mathrm{= 6\pi}\)

6. TRANSLATE back to solve for k

• We know: sector area = \(\mathrm{k\pi}\)
• We found: sector area = \(\mathrm{6\pi}\)
• Therefore: \(\mathrm{k\pi = 6\pi}\)
• Dividing both sides by \(\mathrm{\pi}\): \(\mathrm{k = 6}\)

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that sector area relates to the angle fraction of a full circle

Many students try to use complex sector formulas or get confused about how angles relate to areas. They might attempt to directly use the \(\mathrm{60°}\) measurement without connecting it to the \(\mathrm{360°}\) full circle, leading to incorrect calculations like using \(\mathrm{60\pi}\) or other arbitrary combinations. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors in the fraction calculation

Students correctly identify that they need \(\mathrm{(60/360) \times 36\pi}\) but then make calculation errors. Common mistakes include:

• Calculating 60/360 as 1/3 instead of 1/6
• Computing \(\mathrm{(1/6) \times 36\pi}\) as \(\mathrm{36\pi/6 = 6}\) (forgetting the \(\mathrm{\pi}\))
• These errors lead them to get stuck and randomly select an answer

The Bottom Line:

This problem tests whether students can connect angle measurements to area ratios - a key geometric relationship that bridges angular and area concepts in circle geometry.