# 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 \(19.6\mbox{ cm.}\)