Question:A sector of a circle has a central angle measuring 72°. The area of this sector is 18 square inches....

1. TRANSLATE the problem information

• Given information:
  • Sector has central angle of \(72°\)
  • Sector area = \(18\) square inches
  • Need to find total circle area

2. INFER the key relationship

• A sector's area compared to the whole circle follows the same proportion as its central angle compared to \(360°\)
• This means: \(\frac{\mathrm{Sector\,area}}{\mathrm{Total\,area}} = \frac{\mathrm{Central\,angle}}{360°}\)
• This approach avoids needing to find the radius first

3. TRANSLATE into equation form

Set up the proportion:

\(\frac{18}{\mathrm{Total\,area}} = \frac{72°}{360°}\)

4. SIMPLIFY the right side

• \(\frac{72°}{360°} = \frac{72}{360} = \frac{1}{5}\)
• So: \(\frac{18}{\mathrm{Total\,area}} = \frac{1}{5}\)

5. SIMPLIFY to solve for total area

• Cross multiply: \(18 = \frac{1}{5} \times \mathrm{Total\,area}\)
• Multiply both sides by 5: \(\mathrm{Total\,area} = 18 \times 5 = 90\)

Answer: 90 square inches (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to use the sector area formula \(\mathrm{A} = \frac{θ}{360°} \times πr^2\) but get stuck because they don't know the radius. They may attempt to work backwards to find radius first, making the problem unnecessarily complicated and leading to calculation errors or abandoning the systematic approach altogether.

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the proportion incorrectly, perhaps writing \(\frac{72°}{18} = \frac{360°}{\mathrm{Total\,area}}\), which flips the relationship. This leads them to calculate \(\mathrm{Total\,area} = \frac{360 \times 18}{72} = 90\), which coincidentally gives the right answer, or they might make an arithmetic error and select Choice C (72).

The Bottom Line:

This problem tests whether students recognize the direct proportional relationship between sector area and central angle, rather than getting bogged down in complex formula manipulation.