Arc Length and Sector Area
Arc Length
The length of an arc on a circle of radius is equal to the radius multiplied by the angle subtended by the arc in radians. Using to denote arc length we have
.
This should actually be intuitive since the arc length on the unit circle is equivalent to the angle in radians. The figure below shows arc length between Points and on the circle or radius . Since we are looking at a length, we always consider the angle subtended by and to be positive. (In each of the next two figures, both points and can be moved.)Interactive Arc Length
Area of a Circular Sector
Recall that the area of a circle of radius is given by
.
A circular sector is a wedge made of a portion of a circle based on the central angle (in radians) subtended by an arc on the circle. Since the angle around the entire circle is radians, we can divide the angle of the sector's central angle by the angle of the whole circle 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.