Overview

Arc and chord formulas are useful for circle measurement problems, especially with central angles.

Core Skills

Convert Degrees to Radians

Use $\theta$ in radians for arc length; convert early to avoid mistakes.

Relate Chords and Central Angles

Use $c=2r\sin(\theta/2)$ to connect lengths to angles.

Use Symmetry

Equal chords subtend equal central angles and equal arcs.

Key Ideas

• Arc length: $s = r\theta$ (radians).
• Chord length: $c = 2r\sin(\theta/2)$.

Worked Example

A circle has radius $10$ and central angle $60^\circ$. Find the chord length.

$\theta=\pi/3$, so $c = 2\cdot 10\sin(\pi/6) = 10$.

More Examples

Example 1: Arc Length

Find the arc length when $r=6$ and $\theta=150^\circ$.

$\theta=5\pi/6$, so $s=r\theta=5\pi$.

Example 2: Central Angle from Chord

If $r=5$ and chord length is $5$, find the central angle.

$5=2\cdot 5\sin(\theta/2)$ gives $\sin(\theta/2)=1/2$, so $\theta=60^\circ$.

Example 3: Equal Chords

If two chords have equal length, what can you conclude about their arcs?

They subtend equal arcs and equal central angles.

Strategy Checklist

• Convert to radians for arc length.
• Use $c=2r\sin(\theta/2)$ for chord problems.
• Apply symmetry for equal chords.

Common Pitfalls

• Using degrees in the arc length formula.
• Mixing up arc length and chord length.
• Forgetting the half-angle in the chord formula.

Practice Problems