Arcs & Sectors

TipLearning Objectives
  • Compute arc length and sector area.
  • Use proportional reasoning with central angles.
  • Interpret arcs and sectors in geometric and applied contexts.

Key Ideas

Arcs and sectors are fractions of a circle based on the central angle.

Arc Length

\[ \text{Arc length} = \frac{\theta}{360^\circ} \cdot 2\pi r \]

Sector Area

\[ \text{Sector area} = \frac{\theta}{360^\circ} \cdot \pi r^2 \]

Circle with labeled radius, arc, central angle, and shaded sector.

Common Problem Types

Arc Length from Degrees

Proportion of circumference.

Sector Area from Degrees

Proportion of total area.

Finding Angle Given Arc Length / Area

Solve proportion for \(\theta\).

Real-World Applications

Pizza slices, fan blades, rotation, radar sweep angles.

Strategies

  • Set up proportion:
    \[\frac{\theta}{360} = \frac{\text{part}}{\text{whole}}.\]
  • Keep \(\pi\) exact for cleaner answers.
  • Convert to radians only when required.

Worked Examples

Example 1

In a circle with radius 12, find arc length for a 45° arc.

\[ \frac{45}{360} \cdot 2\pi(12) = \frac{1}{8} \cdot 24\pi = 3\pi. \]

Example 2

Find sector area for radius 10 and central angle 90°.

\[ \frac{90}{360} \cdot \pi(10^2) = \frac14 100\pi = 25\pi. \]

WarningCommon Mistakes
  • Forgetting arc length uses circumference, not area.
  • Using degrees when formula expects radians (or vice versa).
  • Dividing by 180 instead of 360.

Practice Problems

  1. Radius = 6, angle = 60° → arc length.
  2. Radius = 4, angle = 30° → sector area.
  3. Arc length = \(5\pi\), radius = 10. Find angle.
  1. \(=2\pi\sqrt{}? → (60/360)(12\pi)=2\pi.\)
  2. \(=(30/360)(16\pi)=\frac{1}{12}16\pi=\frac43\pi.\)
  3. Solve \(5\pi = (\theta/360)(20\pi)\) → \(\theta=90^\circ\).

Summary

  • Arc length ∝ circumference.
  • Sector area ∝ circle area.
  • Always divide angle by 360.
  • When stuck, rewrite as part/whole proportions.