In a circle, a chord divides the circle into two regions, which are known as the segments of a circle.

The two segments of the circle are known as the minor and the major segment respectively.

The region bounded by the chord is called the minor segment and the minor arc intercepted by the chord. In short, the segment having a smaller area is the minor segment.

The region bounded by the chord is called the major segment and the major arc intercepted by the chord. In short, the segment having a larger area is the major segment.

In the same segment of a circle the angles are equal.

In the figure above, the chord AB divides the circle into two parts, the upper half namely the major segment and the lower half namely the minor segment.

Derivation of The Formula for the Area of Segment of Circle

Theorem:

Let C be a circle with Radius= r.

Let S = segment of a circle C such that the base of the circle subtends an angle = at the centre of the circle.

Then, the (A) area of segment of a circle(S) is given by:

Proof:

Here, let BCDE= Segment(S),

b= Length of the base of the Segment(S),

BACE= Sector of C whose angle =

A= Area of BACE – Area of isosceles ∆ABC with base b.

h= Altitude of ∆ABC

Theorems on Segment of a Circle

The three main theorems based on a circle’s segment are as follows:

### Alternate Segment Theorem:

The other name for the theorem is the tangent-chord theorem.

The theorem states that the angle between the tangent and the side of the triangle is equal to the opposite interior angle.

The diagram given below shows the pair of alternate angles.

### Angle in the same segment theorem:

The theorem states that the angles subtended by the same arc at the circumference of the circle are equal.

We can also say that the angles which are on the same segment in a circle are always equal.

QUESTIONS TO SOLVE:

Question 1) Find the area of segment of circle that has a radius of 8 cm?

Solution-> Let’s list down the given information,

Radius(r) = 8 cm

Angle =180°

The area of segment of circle can be calculated with the formula:

Area of segment= Area of sector- Area of triangle ---- equation (1)

Now, we will find the area of a triangle,

Let’s see the isosceles triangle a bit more closely,

Now we will calculate the area for the triangle,

Question 2) Kaushik eats a slice of pizza which has a radius of 8 inches and forms an angle of 30 degrees at the centre. Find the area of the slice?

Solution-> This slice of pizza is a part of the big circular pizza; therefore, we can say that every slice of pizza is a sector.

Question 3) Find the area of angle formed by the chords with radius 20 cm, if the length of the corresponding arc is 22cm.

Solution-> Let us draw a diagram for better understanding of the given question:

Since we do not know the measure of the angle, we will find the angle,

We know that,

Question 4) The radius of a circle is 10 cm and the given angle is 30 degrees. Find the segment of the circle in radians using the area of segment formula.

Solution-> Let’s list down the information given,

Radius (r) = 10 cm

We know that the area of segment of circle formula is:

Terms and Facts, You Need to Know!

Chord: A chord is basically a line segment that connects two points on a curve. In a circle, the diameter is the longest chord.

A semicircle is the biggest segment of a circle.

FAQ (Frequently Asked Questions)

1. What is the segment of a circle?

In a circle, a chord divides the circle into two regions, which are known as the segments of a circle and two segments of the circle are known as the minor and the major segment respectively.

How do you find the area of a segment of a circle?

Here is the area of segment of circle formula,