Circular sector

Area

The total area of a circle is πr². The area of a sector in terms of L can be obtained by multiplying the total area πr² by the ratio of L to the total perimeter 2πr. $${\displaystyle A=\pi r^2\frac{L}{2\pi r}=\frac{rL}{2}}$$ The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2π (because the area of the sector is directly proportional to its angle, and 2π is the angle for the whole circle, in radians): $${\displaystyle A=\pi r^2\frac{\theta }{2\pi }=\frac{r^2\theta }{2}}$$ Another approach is to consider this area as the result of the following integral: $${\displaystyle A={\int }_0^{\theta }{\int }_0^{r}dA={\int }_0^{\theta }{\int }_0^r\tilde{r}d\tilde{r}d\tilde{\theta }={\int }_0^{\theta }\frac{1}{2}{r}^{2}d\tilde{\theta }=\frac{r^2\theta }{2}}$$

Converting the central angle into degrees gives $${\displaystyle A=\pi r^2\frac{{\theta }^{\circ }}{{360}^{\circ }}}$$

Arc length

The formula for the length of an arc is: $${\displaystyle L=r\theta }$$ where L represents the arc length, r represents the radius of the circle and θ represents the angle in radians made by the arc at the centre of the circle.

If the value of angle is given in degrees, then we can also use the following formula by: $${\displaystyle L=2\pi r\frac{\theta }{{360}^{\circ }}}$$

Perimeter

The length of the perimeter of a sector is the sum of the arc length and the two radii: $${\displaystyle P=L+2r=\theta r+2r=r(\theta +2)}$$ where θ is in radians.

Chord length

The length of a chord formed with the extremal points of the arc is given by $${\displaystyle C=2r\sin \frac{\theta }{2}}$$ where C represents the chord length, r represents the radius of the circle, and θ represents the angular width of the sector in radians.