Sector (Radians)

Statement

For a sector of a circle with radius \(r\) and angle \(\theta\) in radians:

\[ L = r\theta, \quad A = \tfrac{1}{2}r^2\theta \]

where \(L\) is arc length and \(A\) is the sector’s area.

Why it’s true

• The full circumference is \(2\pi r\). For angle \(\theta\), fraction of circle = \(\theta/2\pi\).
• Arc length = fraction × circumference = \((\theta/2\pi)(2\pi r) = r\theta\).
• The full area is \(\pi r^2\). Sector area = fraction × area = \((\theta/2\pi)(\pi r^2)=\tfrac{1}{2}r^2\theta\).

Recipe (how to use it)

1. Ensure angle \(\theta\) is in radians (not degrees).
2. For arc length: multiply radius by angle (\(L=r\theta\)).
3. For sector area: multiply half radius squared by angle (\(A=\tfrac{1}{2}r^2\theta\)).

Spotting it

Look for problems about arc length or area with angles given in radians, e.g. “Find the arc length when radius=5 and angle=2 radians.”

Common pairings

• Radians conversion (degrees ↔ radians).
• Trigonometry problems with circular measure.
• Exam geometry questions involving sectors and arcs.

Mini examples

1. Arc length for \(r=6\), \(\theta=2\) rad → \(L=6×2=12\).
2. Sector area for \(r=4\), \(\theta=\pi/3\) rad → \(A=\tfrac{1}{2}×16×\pi/3=8\pi/3\).
3. Arc length for \(r=10\), \(\theta=\pi/2\) rad → \(L=10×\pi/2=5\pi\).

Pitfalls

• Using degrees directly instead of converting to radians.
• Confusing arc length and area formulas.
• Forgetting to include \(\tfrac{1}{2}\) in the area formula.

Exam strategy

• Always check if the angle is in radians before using formulas.
• Convert degrees to radians: multiply by \(\pi/180\).
• Leave answers in terms of \(\pi\) when exact values are expected.

Summary

For a sector: arc length \(L=r\theta\) and area \(A=\tfrac{1}{2}r^2\theta\), provided \(\theta\) is in radians.