Sectors, segments and degree measure

A chord joins two points on a circle. An arc is part of the circumference. A sector is bounded by two radii and an arc; a segment is bounded by a chord and its arc. The major arc or region is the larger one.

The smaller sector \(OAB\) includes triangle \(OAB\) and the small segment between chord \(AB\) and its arc. If the central angle is \(\alpha^\circ\), it represents the fraction \(\dfrac\alpha{360}\) of a full circle. Thus arc length \(l=\dfrac\alpha{360}(2\pi r)\) and sector area \(A=\dfrac\alpha{360}(\pi r^2)\).

Example

For radius 9 cm and central angle \(80^\circ\), the minor arc length is \(4\pi\text{ cm}\) and the minor sector area is \(18\pi\text{ cm}^2\). The major angle is \(280^\circ\), giving major arc length \(14\pi\text{ cm}\).

A sector’s perimeter includes its two radii: \(2r+l\). An arc length alone is not the perimeter of the sector.