In a circle, a central angle intercepts an arc that is one-sixth of the circle's circumference. What is the measure...

1. TRANSLATE the problem information

• Given information:
  • Central angle intercepts an arc that is one-sixth of the circle's circumference
  • Need to find the central angle in radians

• What this tells us: The arc length is (1/6) × (circumference)

2. INFER the approach

• We can use the arc length formula \(\mathrm{s = r\theta}\) where \(\mathrm{\theta}\) is the central angle in radians
• Since we know the arc is a fraction of the total circumference, we can find the arc length and then solve for \(\mathrm{\theta}\)

3. TRANSLATE the arc length into mathematical terms

• Full circumference = \(\mathrm{2\pi r}\)
• Arc length = \(\mathrm{(1/6)(2\pi r) = \pi r/3}\)

4. SIMPLIFY using the arc length formula

• From \(\mathrm{s = r\theta}\), we have:

\(\mathrm{\pi r/3 = r\theta}\)

• Divide both sides by r:

\(\mathrm{\theta = \pi r/3 \div r = \pi/3}\)

Answer: C) \(\mathrm{\pi/3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "one-sixth of the circle's circumference" and think it means the central angle is \(\mathrm{\pi/6}\) radians, confusing the fraction (1/6) with the angle measure itself.

This reasoning leads them to think: "One-sixth... so the angle must be \(\mathrm{\pi/6}\)."
This may lead them to select Choice A (\(\mathrm{\pi/6}\)).

Second Most Common Error:

Missing conceptual knowledge: Students may not remember that a full circle is \(\mathrm{2\pi}\) radians, instead thinking it's \(\mathrm{\pi}\) radians (confusing it with a semicircle).

Using this incorrect relationship: \(\mathrm{(1/6) \times \pi = \pi/6}\)
This may also lead them to select Choice A (\(\mathrm{\pi/6}\)).

The Bottom Line:

This problem requires careful translation of the English description into mathematical terms and proper application of the arc length formula or proportional reasoning. The key insight is recognizing that "one-sixth of the circumference" describes the arc length, not the angle measure directly.