Segment of a Circle

A segment of a circle is the region that is bounded by an arc and a chord of the circle. When something is divided into parts, each part is referred to as a segment. In the same way, a segment is a part of the circle. But a segment is not any random part of a circle, instead, it is a specific part of a circle that is cut by a chord of it. The segment of circle is the part that is formed by a chord of the circle (intersecting line) and an arc of the circle (part of the boundary).

In this article, we will discuss the concept of segment of circle, and understand its definition and properties. We will learn to find the area and perimeter of the segment of a circle and describe the theorems based on the segment along with some solved examples for a better understanding of the concept.

A segment of a circle is the region that is bounded by an arc and a chord of the circle. Let us recall what is meant by an arc and a chord of the circle.

• An arc is a portion of the circle's circumference.
• A chord is a line segment that joins any two points on the circle's circumference.

There are two types of segments, one is a minor segment, and the other is a major segment. A minor segment is made by a minor arc and a major segment is made by a major arc of the circle.

The properties of a segment of a circle are:

• It is the area that is enclosed by a chord and an arc.
• The angle subtended by the segment at the center of the circle is the same as the angle subtended by the corresponding arc. This angle is usually known as the central angle.
• A minor segment is obtained by removing the corresponding major segment from the total area of the circle.
• A major segment is obtained by removing the corresponding minor segment from the total area of the circle.
• A semicircle is the largest segment in any circle formed by the diameter and the corresponding arc.

An arc and two radii of a circle form a sector. These two radii and the chord of the segment together form a triangle. Thus, the area of a segment of a circle is obtained by subtracting the area of the triangle from the area of the sector. i.e., Area of a segment of circle = area of the sector - area of the triangle

Let us use this logic to derive the formulas to find the area of a segment of a circle. Note that this is the area of the minor segment. Usually, a segment of a circle refers to a minor segment.

Note: To find the area of the major segment of a circle, we just subtract the corresponding area of the minor segment from the total area of the circle.

Let us consider the minor segment of the above circle that is made by the chord PQ of a circle of radius 'r' that is centered at 'O'. We know that every arc of a circle subtends an angle at the center which is referred to as the central angle of the arc. The angle made by the arc PQ is θ. We know from trigonometry that, the area of the triangle OPQ is (1/2) r² sin θ. Also, we know that the area of the sector OPQ is:

• (θ / 360°) × πr², if 'θ' is in degrees
• (1/2) × r²θ, if θ' is in radians

Thus, the area of the minor segment of the circle is:

• (θ / 360°) × πr² - (1/2) r² sin θ (OR) r² [πθ/360° - sinθ / 2], if 'θ' is in degrees
• (1/2) × r²θ - (1/2) r² sin θ (OR) (r² / 2) [θ - sin θ], if 'θ' is in radians

We know that the segment of a circle is made up of an arc and a chord of the circle. Consider the same segment as in the above figure.

Perimeter of the segment = length of the arc + length of the chord

We know that,

• the length of the arc is rθ, if 'θ' is in radians and πrθ/180, if 'θ' is in degrees.
• the length of the chord = 2r sin (θ/2)

Thus, the perimeter of the segment formula is:

• The perimeter of the segment of a circle = rθ + 2r sin (θ/2), if 'θ' is in radians.
• The perimeter of the segment of a circle = πrθ/180 + 2r sin (θ/2), if 'θ' is in radians.

Mainly, there are two theorems based on the segment of a Circle.

• Angles in the same segment theorem
• Alternate segment theorem

Angles in the Same Segment Theorem

It states that angles formed in the same segment of a circle are always equal.

Alternate Segment Theorem

This theorem states that the angle formed by the tangent and the chord at the point of contact is equal to the angle formed in the alternate segment on the circumference of the circle through the endpoints of the chord.

Important Notes on Segment of Circle

• A segment of circle is the area enclosed by an arc and chord of the circle.
• We have two types of segments of circle - minor and major segment.
• We can find the area of segment using, Area of Segment = Area of Sector - Area of Triangle

Related Articles

• Chord of a Circle
• Geometry
• Circle Formulas

What Is a Segment of a Circle?

A segment of a circle is the region that is bounded by an arc and a chord of the circle. There are two types of segments, one is a minor segment (made by a minor arc) and the other is a major segment (made by a major arc).

What Is the Difference Between Chord and Segment of a Circle?

A chord of a circle is a line segment that joins any two points on its circumference whereas a segment is a region bounded by a chord and an arc of the circle.

What Is the Difference Between Arc and Segment of a Circle?

An arc is a portion of a circle's circumference whereas a segment of a circle is a region bounded by an arc and a chord of the circle.

What Is the Difference Between a Sector of a Circle and a Segment of a Circle?

A sector of a circle is the region enclosed by two radii and the corresponding arc, while a segment of a circle is the region enclosed by a chord and the corresponding arc.

What Is the Formula for Area of the Segment of a Circle?

The area of the segment of the circle (or) minor segment of a circle is:

• (θ / 360°) × πr² - (1/2) r² sin θ (OR) r² [πθ/360° - sin θ/2], if 'θ' is in degrees
• (1/2) × r²θ - (1/2) r² sin θ (OR) (r² / 2) [θ - sin θ], if 'θ' is in radians

Here, 'r' is the radius of the circle and 'θ' is the angle subtended by the arc of the segment.

How To Find the Area of a Segment of a Circle?

Here are the steps to find the area of a segment of a circle.

• Identify the radius of the circle and label it 'r'.
• Identify the central angle made by the arc of the segment and label it 'θ'.
• Find the area of the triangle using the formula (1/2) r² sin θ.
• Find the area of the sector using the formula
  (θ / 360^o) × πr², if 'θ' is in degrees (or)
  (1/2) × r²θ, if θ' is in radians
• Subtract the area of the triangle from the area of the sector to find the area of the segment.

How To Find the Area of a Major Segment of a Circle?

The area of a major segment of a circle is found by subtracting the area of the corresponding minor segment from the total area of the circle.

What Is the Alternate Segment Theorem of a Circle?

The alternate segment theorem states that the angle formed by the tangent and the chord at the point of contact is equal to the angle formed in the alternate segment on the circumference of the circle through the endpoints of the chord.

Is a Semicircle a Segment of the Circle?

We know that a diameter of a circle is also a chord of the circle (in fact, it is the longest chord of the circle). Also, we know that the semicircle's circumference is an arc of the circle. Thus, a semicircle is bounded by a chord and an arc and hence is a segment of the circle.

Are the Angles in the Same Segment of a Circle Equal?

Yes, the angles formed by the same segment of a circle are equal. i.e., the angles on the circumference of the circle made by the same arc are equal.

How to Find the Perimeter of Segment of Circle?

The perimeter of a segment of circle can be calculated by adding the length of the chord of circle and the length of the corresponding arc of the circle. The formula for the perimeter of segement is 2r sin (θ/2) + rθ.