A circle has a radius of 18. A central angle in this circle subtends an arc with a length of...

1. TRANSLATE the problem information

• Given information:
  • Circle radius: \(\mathrm{r = 18}\)
  • Arc length: \(\mathrm{s = 2π}\)
  • Need to find: central angle in degrees

2. INFER the approach

• We need to use the arc length formula that connects these three quantities
• The key formula is \(\mathrm{s = rθ}\) where \(\mathrm{θ}\) must be in radians
• Since answer choices are in degrees, we'll need to convert our final answer

3. SIMPLIFY to find the angle in radians

• Substitute into \(\mathrm{s = rθ}\):

\(\mathrm{2π = 18θ}\)

• Solve for \(\mathrm{θ}\):

\(\mathrm{θ = \frac{2π}{18} = \frac{π}{9}}\) radians

4. SIMPLIFY to convert from radians to degrees

• Use conversion factor: multiply radians by \(\mathrm{\frac{180°}{π}}\)
• \(\mathrm{θ}\) in degrees = \(\mathrm{\frac{π}{9} × \frac{180°}{π}}\)
• The \(\mathrm{π}\) terms cancel: \(\mathrm{θ = \frac{180°}{9} = 20°}\)

Answer: B. 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Arc length formula
Students may not remember that \(\mathrm{s = rθ}\) requires \(\mathrm{θ}\) to be in radians, or they might confuse it with other circle formulas like circumference. Without this foundational formula, they cannot set up the problem correctly and may resort to guessing or using incorrect relationships.
This leads to confusion and guessing.

Second Most Common Error:

Weak INFER reasoning: Forgetting unit conversion
Students correctly find \(\mathrm{θ = \frac{π}{9}}\) radians but fail to recognize that the answer choices are in degrees. They might try to match \(\mathrm{\frac{π}{9} ≈ 0.35}\) to one of the given options or convert incorrectly.
This may lead them to select Choice A (10) if they approximate \(\mathrm{\frac{π}{9}}\) and make conversion errors.

The Bottom Line:

This problem tests the essential connection between arc length and central angles, requiring both formula recall and careful attention to units. The conversion from radians to degrees is where many students stumble, even after correctly setting up the initial equation.