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Radian Measure

Arc Length

The arc length can be found using the formula
\[ \mbox{Arc Length} = \theta r \] where \(\theta\) is measured in radians and \(r\) is the radius of the circle.

Example: for the diagram below, calculate the radius of the sector, given that the length of the arc is \(12\) cm and the angle subtended by the arc is \(35^{\circ}\).

 

To find the radius, we firstly need to convert the angle to be in radians:
\[ 35^{\circ} = \frac{\pi}{180} \times 35 = \frac{7\pi}{36}\] 
Secondly, we need to rearrange the arc length formula to make the radius the subject:
\begin{eqnarray*}
\mbox{Arc Length} &=& \theta r \\
r &=& \frac{\mbox{Arc Length}}{\theta} 
\end{eqnarray*}
Finally, substitute in the given information:
\begin{eqnarray*}
r &=& \frac{\mbox{Arc Length}}{\theta} \\
&=& \frac{12}{\frac{7\pi}{36}} \\
&\approx& 19.6\mbox{ cm}
\end{eqnarray*}
Therefore, the radius is approximately \(19.6\mbox{ cm.}\)