Radians, Arcs and Sectors

One radian subtends an arc equal in length to the radius. \(180^\circ=\pi\) radians; multiply degrees by \(\dfrac{\pi}{180}\) to convert to radians.

For \(\theta\) in radians, \(s=r\theta\) and \(A_{\text{sector}}=\dfrac12r^2\theta\). Sector perimeter is \(r\theta+2r\), not just arc length.

For a minor segment with \(0\le\theta\le\pi\), subtract the triangle: \(A_{\text{segment}}=\dfrac12r^2(\theta-\sin\theta)\). Select the intended minor or major region before calculating.

Example

For \(r=6\), \(\theta=\dfrac{2\pi}{3}\): \(s=4\pi\), sector area \(12\pi\), and segment area \(12\pi-9\sqrt3\).

A chord joins two points on the circumference. A sector is bounded by two radii and an arc; a segment by a chord and an arc. Convert radians to degrees by multiplying by \(180/\pi\). For a major sector use the reflex angle \(2\pi-\theta\). Trace the actual boundary: internal lines do not contribute to perimeter.

Two concentric sectors with radii \(R>r>0\) and the same angle \(0<\theta<2\pi\) form an annular sector. Its area is \(\dfrac12(R^2-r^2)\theta\), and its perimeter is \((R+r)\theta+2(R-r)\). A major segment is the whole circle with its corresponding minor segment removed. Use radians, length units for perimeter and square units for area.