Area of Segment of a Circle

Definition of Circle

A circle is a closed, two-dimensional shape formed by joining a set of points drawn equidistant from a certain point, ‘the centre’. The line of reflection symmetry is formed by every line passing through the circle. In addition, at every angle, it exhibits rotary symmetry around the center.

Definition of Segment

The area bordered by an arc and a circle chord is one segment of a circle. Each portion is called a segment when anything is split up into pieces. A segment is not a random part of a circle; instead, it is a certain part of a circle split by a chord.

• Arc: A minute portion of the circumference of a circle is known as an arc.
• Chord: A chord is a line segment inscribed inside the circle between 2 points. It connects any two points drawn on the circumference of the circle.

Types of Segments

There are two types of segments present in a circle, namely the Minor segment and the Major Segment. Let us assume that a circle is divided by a chord. Therefore, the segment formed by the minor arc is termed as the minor segment, and likewise, the segment formed by the major arc is known as the major segment of the circle.

Area of Segment of a Circle

Let us derive the formula and the concept for determining the area of the segment within a given circle.

• First, let us draw a circle with radius ‘r’ units.
• Then let us draw a line segment AB as shown in the figure. Let the center point of the circle be O.
• Therefore, the radius from point O to point A and B will form an isosceles triangle OAB.
• Assuming the angle extended by the radii on the center to be θ (theta). From the figure, we can deduce that the area of the segment is equal to the difference between the area of sector AOB and the area of triangle OAB.

Area of minor segment = Area (sector AOB) – Area (isosceles triangle OAB)

Formula:

Area of triangle OAB: We are provided with two defined lengths of the triangle OAB, OA and OB are the radii of the circle and the angle ‘θ’ extended by these lengths. According to the formula of the area of the isosceles triangle, the area comes out to be equal to ½ x r² x sin θ.

Area of Sector AOB: The formula for the sector of a circle in degrees is θ/360 x pi x r² and ½ x r² x θ in radians.

Therefore, the area of the segment in

Degrees: θ/360 x pi x r² - ½ x r² x sin θ = r2 {(pi x θ)/360 – sin θ/2}

Radians: ½ x r² x θ - ½ x r² x sin θ = r² {(θ – sin θ)/2}

Note: We can also find the area of the major segment by subtracting the area of the minor segment from the area of the whole circle.