An angle has a measure of 9pi/20 radians. What is the measure of the angle in degrees?

1. TRANSLATE the problem information

• Given information:
  • Angle measure: \(\frac{9\pi}{20}\) radians
  • Need to find: measure in degrees

• This tells us we need to convert from radians to degrees using the conversion formula

2. INFER the approach

• To convert radians to degrees, multiply by the conversion factor \(\frac{180°}{\pi}\)
• This conversion factor comes from the relationship that \(\pi \text{ radians} = 180°\)
• Set up: \(\frac{9\pi}{20} \times \frac{180°}{\pi}\)

3. SIMPLIFY the expression

• Multiply the fractions: \(\frac{9\pi}{20} \times \frac{180°}{\pi} = \frac{9\pi \times 180°}{20 \times \pi}\)
• Cancel \(\pi\) from numerator and denominator: \(\frac{9 \times 180°}{20}\)
• Calculate: \(9 \times 180 = 1620\), so we have \(\frac{1620°}{20}\)
• Divide: \(1620 \div 20 = 81\)

Answer: 81 degrees

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating \(9 \times 180\) or when dividing \(1620\) by \(20\).

For example, they might calculate \(9 \times 180\) as \(1800\) instead of \(1620\), leading to \(\frac{1800}{20} = 90\) degrees instead of the correct 81 degrees. Or they might divide incorrectly, getting \(\frac{1620}{20} = 80\) instead of 81.

Second Most Common Error:

Poor TRANSLATE reasoning: Students use the wrong conversion formula, applying the degree-to-radian conversion (multiply by \(\frac{\pi}{180°}\)) instead of the radian-to-degree conversion.

This leads them to calculate \(\frac{9\pi}{20} \times \frac{\pi}{180°}\), which gives a much smaller result and doesn't make sense dimensionally since they end up with \(\frac{\pi^2}{400}\) instead of a degree measure.

The Bottom Line:

This problem tests whether students can correctly apply the radian-to-degree conversion formula and perform accurate arithmetic. The key insight is recognizing that converting FROM radians TO degrees requires multiplying by \(\frac{180°}{\pi}\), not \(\frac{\pi}{180°}\).