Area of Segment of a Circle: Types, Formulas, Theorems, Examples

Area of segment of a circle is obtained through subtracting the area of the triangle from the area of the sector. A line segment that joins two points on a circle or curve with both the end points lying on the arc is called a Chord. The region of the circle cut off from the rest of the circle by this chord is called as segment of a circle. This segment is a part of a circle in two dimensional space bounded by a chord whose end points lie on the arc. In other words, the segments can be defined as the parts that are divided by the circle's arc and connected with the chord through its end points.

Keyterms: Circle, Traingle, Line segment, Chord, Major segment, Minor segment, Radians, Degree

Also Read: Surface Areas and Volumes

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Types of Segments in a Circle

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The segments of a circle can be classified into two types that are Major segment and Minor segment. When the chord divides the regions of a circle into segments, the segment that has larger area is known as a major segment and the segment that has smaller area is known as Minor segment.

The major segment consists of a major arc and the minor segment consists of a minor arc. Here it can be observed that ADC is the major segment and ABC is the minor segment. Besides, the area of these segments of a circle can also be calculated using formulas.

Types of Segments in a Circle

Also, read: Areas Related to Circles Revision Notes

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Area of Segment of a Circle: Formula

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The formula that can be used to calculate the area of segments of a circle is as follows. It can be calculated either in terms of degree or radian.

• Area of Segment in Radians: A= (½) × r^2 (θ – Sin θ)
• Area of Segment in Degree: A= (½) × r^ 2 × [(π/180) θ – sin θ]

Derivation

Let us consider a circle which has a triangle AOB circumscribed within. 

Let  ∠AOB = θ° 

And area of triangle AOB is AΔAOB

Then the Area of sector AOBC = θ/360° × πr² (Formula).

So, the area of Segment of Circle can be calculated as

Area of Segment = Area of Sector – Area of Triangle

Sector of a Circle

A = θ/360° × πr² – A ΔAOB

To find the area of triangle AOB we need to calculate the sides.

Using Pythagoras theorem  

OP =\({\sqrt{[r^2 -(\frac{AB}{2}) 2]}}\) in case the length of AB is given.

Or OP= rcos(θ/2) in case the angle θ is given.

Area of triangle= ½ × Base × Height

= ½ × AB × OP.

In this way the area of Segment of a circle can be calculated by substituting the given values. 

Area of Segment in Degree: A= (½) × r^ 2 × [(π/180) θ – sin θ]

Sector vs Segment of a Circle

Also Read:

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Theorems on Segment of Circle

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There are two theorems on segments of a circle. 

Alternate Segment Theorem

The theorem states that in a circle, the angle which lies between the chord and tangent passing through the end points is equal to the angle in the alternate segment. 

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Here the angle lying between the tangent DE and chord BC is ∠BCE and the angle in alternate segment is ∠BAC.

According to the theorem, ∠BCE = ∠BAC.

Another angle lying between tangent DE and chord AC is ∠ACD and the angle in alternate segment is ∠ABC.

According to the theorem, ∠ACD = ∠ABC.

Angle in the same Segment Theorem

The theorem states that the angles formed in a segment are all equal i.e., angles in the same segment of a circle are equal.

According to the theorem, ∠PAQ = ∠PBQ.

From the inscribed angle Theorem, it can be inferred that an angle formed by an arc at the center of a circle is double the angle divided by the same arc at any other point in the circle.

Therefore, ∠POQ = 2∠PAQ

∠POQ = 2∠PBQ.

From these two we can conclude that 

2∠PAQ= 2∠PBQ

Hence, ∠PAQ = ∠PBQ

Read more Important Questions on Circles

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Things to Remember

• The region of the circle cut off from the rest of the circle by this chord is called a Segment of a circle.
• It is a part of a circle in two dimensional space bounded by a chord that has its end points lying on the arc.
• There are two types of Segments: Major segment and Minor segment.
• The segment that has a larger area is called the Major segment and the segment that has smaller area is called the minor segment.
• The formula that can be used to calculate Area of Segment in Degree: A= (½) × r^ 2 × [(π/180) θ – sin θ]
• There are two theorems on Segment of Circle that are Alternate segment theorem and Angle in the same segment theorem.
• Alternate Segment Theorem states that in a circle, the angle which lies between the chord and tangent passing through the end points is equal to the angle in the alternate segment. 
• Angle in the same segment Theorem states that the angles formed in a segment are all equal i.e., angles in the same segment of a circle are equal.

Also, read: Circles Revision Notes