The segment of a circle is a part of circle that is obtained when a circle and a chord (or a secant) intersect each other.
When a chord or a secant is drawn through a circle, it is divided into two portions which are known as segments of circle. The bigger segment of a circle is called major segment, while the smaller one is known as minor segment as shown in the following figure:

Circular Segmentation

Above diagram also shows the major and minor arcs of a circle. The part of circumference of circle corresponding to major segment is termed as major arc and that corresponding to minor segment is known as minor arc.
The process of dividing a circle into segments by means of a chord or a secant, is called circular segmentation.There are many problems and applications in various fields based on circular segmentation and finding its area.

If a chord divides a circle in two segments, then either of the segments subtends an angle on the center which is referred as central angle. In order to calculate the area of a segment, we first need to know the area of sector (pizza-slice shaped figure in a circle) made on the corresponding segment.
Let us have a look at the diagram illustrated below :
 Area of Circular Segment
In this diagram, ABS is a segment and AOBS is the corresponding sector. Then area of segment ABS can be calculated by subtracting area of $\bigtriangleup$AOB from area of sector AOBS.

Also, the area of the major segment so obtained, can be determined by subtracting area of segment ABS from the area of full circle.
We can say that
Area of segment = Area of sector - Area of corresponding triangle
Let us consider a circle with radius r. Let there be a sector making a central angle $\theta$.
Then, the area of sector = $\pi r^{2}.$ $\frac{\theta}{2 \pi}$ = $\frac {1}{2}$ $\times$ $ r^{2} \theta$
(when $\theta$ is in radians)

Area of triangle = $r.$$\frac{1}{2}$$rsin \theta$ = $\frac{1}{2}$$r^{2} sin \theta$

Area of segment = $\frac {1}{2}$$ r^{2} \theta-$$\frac{1}{2}$$r^{2}$$ sin \theta$ Hence,
area of segment (in radians) is given by the formula:
Area of segment = $\frac {1}{2}$ $ r^{2} (\theta-sin \theta)$
area of segment (in degrees) is determined by the following formula:

Area of segment = $\frac {1}{2}$ $r^{2} (\theta$ $\frac{\pi}{180}$ $-sin \theta)$

Length of segment is represented by the arc length of segment, i.e. length of segment is referred to the length of the arc formed by the segment.
Arc Length
The formula for arc length is given by the following relation:
Arc length of segment = Radius $\times$ Angle subtended by the arc
If the radius of circle is denoted by "r" and the angle subtended by the segment or the corresponding arc be $\theta$, then length of segment, when $\theta$ is calculated in radians, is defined by the following formula:
L = r $\times$ $\theta$
And, the length of segment, when $\theta$ is given in degrees, is determined using the formula given below:
L = r $\times$ $\theta$$\frac{\pi}{180}$

Few problems based on area of segment of a circle are illustrated below:

Problem 1: Calculate the area of a segment of a circle whose radius is 6 cm. Provided that segment subtends an angle of 30$^{\circ}$ at center.
Given that:
r = 6 cm
$\theta = 30^{\circ}$ = $30^{\circ} \times$$ \frac{\pi}{180}$

$\theta$ =$\frac{\pi}{6}$

The formula for area of segment of a circle is given by:

Area of segment = $\frac {1}{2}$ $r^{2} (\theta-sin \theta)$

Area of segment = $\frac {1}{2}$$\times 6^{2} $($\frac{\pi}{6}$-$sin$ $\frac{\pi}{6}$)

= 18 ($\frac{\pi}{6}$-$\frac{1}{2}$)

= $\frac{3 \times 22}{7}$-9

= 0.429 cm$^{2}$

Problem 2: If the area of a segment of a circle which makes an angle of 120$^{\circ}$ at center, is 12 cm$^{2}$. Find the radius of circle.
Given that:
$\theta$ = 120$^{\circ}$

$\theta$ = 120 $\times$ $\frac{\pi}{180}$ = $\frac{2}{3} \pi$

Area of segment = $\frac {1}{2}$$ r^{2}$$ (\theta-sin \theta)$

$12$=$\frac {1}{2}$$ r^{2}$($\frac{2}{3}$$ \pi$-$sin$($\frac{2}{3}$$ \pi$))

$12$=$\frac {1}{2}$$ r^{2}$($\frac{2}{3}$$\pi$- $\frac{\sqrt{3}}{2}$)

$24$ = r$^{2} (2.095- 0.866)$

$r^{2}$ =$\frac{24}{1.229}$

r = 4.42 cm (approx)