26.13: Arc Length
- Page ID
- 174414
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Definitions and Theorems
An arc along a circle is a portion of the circumference of a circle. It's the curved line connecting two points on the circle's edge. The length of this curve is called the arc length.
An angle with vertex at the center of a circle is called a central angle of the circle.
A central angle whose initial and terminal sides meet the endpoints of an arc is said to subtend (or span) the arc.
The length of the arc, \( s \), in a circle of radius \( r \), spanned by a fraction of a revolution, \( p \), is\[s = p \cdot(2 \pi r). \nonumber \]
On a circle of radius \(r\), the length \(s\) of an arc spanned by an angle \(\theta\) in radians is\[s=r \theta. \nonumber \]
The radian measure of an angle is given by\[ \text{fraction of one revolution } \times 2 \pi . \nonumber \]
\[\begin{array}{rccccl}
& 2 \pi \text{ radians} & = & 360^{\circ} & & \\[6pt]
\implies & 1 & = & \dfrac{180^{\circ}}{\pi \text{ radians}} & \quad & (\text{dividing both sides by }2 \pi\text{ radians}) \\[6pt]
& & \text{and} & & & \\[6pt]
\implies & \dfrac{\pi \text{ radians}}{180^{ \circ }} & = & 1 & \quad & (\text{dividing both sides instead by } 360^{\circ}) \\[6pt]
\end{array} \nonumber \]