Segment of a Circle: Definitions, Types, Area with Formula, How to Calculate, Theorems, and Properties

Segment of a Circle Overview

A segment is the area of a circle between the chord and the arc. A circle is a route that a point equidistant from a single point on the plane may follow; this point is known as the circle's center, and the distance between it and that point is known as the circle's radius. A segment is a section of a circle's interior. A sector is the region that a segment encloses and the angle that a segment occupies. By deducting the triangle produced inside the sector from the sector that contains the segment, one may calculate the area of a circular segment. 

What is a Circle?

A circle is a shape formed by all points on a plane that is at a particular distance from the center. The radius is the distance between any two points on a circle and the center. The circle has been known since before written history began. Natural circles, such as the full moon or a slice of round fruit, are frequent. The circle is the foundation for the wheel, which, together with related developments like gears, allows for the creation of much contemporary technology. The study of the circle has influenced the development of geometry, astronomy, and calculus in mathematics.

What is the Segment of a Circle?

A region enclosed by a chord and a matching arc located between the chord's ends is referred to as a segment of a circle. To put it another way, a circular segment is a section of a circle that divides from the remainder of the circle along a secant or chord. Segments are also the components that the arc of the circle divides into and connects to its ends through a chord. The center point is absent from the segments, it should be observed.

Types of Segment of a Circle

A segment of the circle is, by definition, the portion of a circle that is encompassed by a chord and its matching arc. The main segment and the minor segment are the two categories for segments of a circle. The major segment is the one with the bigger size, while the minor segment is the one with the lesser area.

Read more about the Father of Mathematics, Sphere Formula, and Signum Function.

Area of Segment of a Circle Formula 

Either radians or degrees can be used in the calculation to calculate segment area. The following are the formulae for a circle's segment-

Theorems of Segment of a Circle

Based on the segments of a circle, there are two fundamental theorems-

Alternate Segment Theorem

According to this theorem, the angle created at the point of contact between the tangent and the chord is equal to the angle created by the alternate segment on the circle's circumference via the chord's ends.

Angles in the Same Segment Theorem

It asserts that angles created in the same circle segment are always equal.

Read more about the Area of a Parallelogram and the Area of Square.

Properties of  Segment of a Circle

A circle segment's characteristics are as follows-

1. Any circle's biggest section, created by its diameter and associated arc, is known as a semicircle.
2. By subtracting the appropriate minor segment from the circle's overall area, a major segment is obtained.
3. By subtracting the matching major segment from the circle's overall area, a minor segment is created.
4. It is the region that is bounded by an arc and a chord.
5. The arc's matching segment also has the same angle as the segment at the centre of the circle. Typically, this angle is referred to as the centre angle.

How to solve problems involving a Segment of a Circle?

In order to resolve issues concerning a circle segment-

Questions about the Area

1. Determine the radius' length.
2. Measure the sector's angle to determine its size.
3. Discover the sector's region.
4. Calculate the area of the triangle formed by a circle's radii and chord.
5. Subtract the triangle's area from the sector's area.

Questions about the Perimeter

1. Determine the radius' length. 
2. Measure the sector's angle to determine its size. 
3. Determine the circle segment's arc's length.
4. Calculate the segment's chord length.
5. Add the chord's length and the arc's length.

Read more about the Sphere Formula.

Area of a Segment of a Circle

A sector is made up of an arc and two circle radii. Together, these two radii plus the segment's chord make a triangle. As a result, the area of a circle's segment is calculated by deducting the triangle's area from the sector's area. i.e., the Area of a circle segment is equal to the sum of its sectors and triangles. Remember that the minor segment is in this place. A circle's minor portion is typically referred to as its segment.

Points to Remember

1. The region bounded by a circle's arc and chord is known as a segment.
2. There are two different kinds of circle segments: minor and major segments.
3. Using this, we can determine the segment's area. Area of Triangle - The Area of the Sector gives the area of a segment.