[justify]The length of an arc on a circle of radius [math]r[/math] is equal to the radius multiplied by the angle subtended by the arc in radians. Using [math]s[/math] to denote arc length we have [/justify][center][math]\Large s=r \theta[/math].[/center][br]This should actually be intuitive since the arc length on the unit circle is equivalent to the angle in radians. [br][br]The figure below shows arc length between Points [math]A[/math] and [math]B[/math] on the circle or radius [math]r[/math]. Since we are looking at a length, we always consider the angle [math]\theta [/math] subtended by [math]A[/math] and [math]B[/math] to be positive. (In each of the next two figures, both points [math] A [/math] and [math] B [/math] can be moved.)

[justify]Recall that the area [math]A[/math] of a circle of radius [math]r[/math] is given by[/justify][center][math]A = \pi r^2[/math].[/center][br][justify]A circular sector is a wedge made of a portion of a circle based on the central angle [math]\theta[/math] (in radians) subtended by an arc on the circle. Since the angle around the entire circle is [math]2 \pi[/math] radians, we can divide the angle of the sector's central angle by the angle of the whole circle [math](2 \pi)[/math] to determine the fraction of the circle we are solving for. Then multiply by the area of the whole circle to derive the sector area formula.[/justify][br][br][center][math]\begin{align} A = & \frac{\theta}{2 \pi}\cdot \pi r^2 \\ \\ A = & \frac{\theta r^2}{2} \end{align} [/math][/center]