Segment of a Circle is one of the important parts of the circle other than the sector. As we know, the circle is a 2-D shape in which points are equidistant from the point and the line connecting the two points lying on the circumference of the circle is called the chord of the circle.
The area formed on both sides of this chord is called segment which is the topic of this article. In this article, we will learn about the segments of a circle, its types, and theorems related to it as well. So let's start learning about a segment of a circle.
What is Segment of a Circle?
Segment of a circle is the area bounded by the chord and the arc formed from the endpoint of the chord. In other words, the chords divide the circle into two parts and these parts are called the segment of the circle.
Segment Definition
Segment is formed between the chord and the arc between the endpoint of the chord. A segment can also be defined as the region obtained by subtracting the area of a triangle from the area of the sector of a circle.
Types of Segments
There are two types of segments that are:
- Major Segment
- Minor Segment
Let's discuss these types in detail as follows:
Major Segment
The segment which is bigger in a circle is called the major segment. The segment with the greater area is called the major segment.
Minor Segment
The segment which is smaller in a circle is called the minor segment. The segment with the smaller area is called the minor segment.
Special Case: Semicirlce
Other than these two types, there is one more special case of segments when both the segments become an equation i.e., semicircle. Semicircle is the largest segment in a circle as the diameter is the largest chord of the circle.
There are two important formulas related to the segment of a circle
- Area of Segment of a Circle
- Perimeter of Segment of a Circle
Let's discuss these formulas as follows:
Area of Segment of a Circle
The area of the segment is the difference between the area of the sector and the area of the triangle. In the above figure area of the segment is determined by subtracting the area of the triangle from the area of the sector.
Area of the segment = Area of the sector - Area of the triangle
Formula for Area of Segment of a Circle
- Area of segment (when θ in radians) = (1/2) × r2(θ - sinθ)
- Area of segment (when θ in degrees) = (1/2) × r2 [(π/180) θ - sinθ]
Perimeter of Segment of a Circle
The perimeter of the segment is the sum of the length of the chord and the length of the arc.
Perimeter of Segment = Length of Chord + Length of the Arc
Formula for Perimeter of Segment of a Circle
- Perimeter of segment (when θ in radians) = rθ + 2rsin(θ/2)
- Perimeter of segment (when θ in degrees) = rθ(π/180) + 2rsin(θ/2)
Theorems on Segment of a Circle
- Alternate Segment Theorem
- Angles in Same Segment Theorem
Alternate Segment Theorem
The alternate segment theorem states that the angle formed by the point of contact of the chord and tangent is equal to the angle formed by the chord of the alternate segment. (angle a = angle b)
Angles in the Same Segment Theorem
Angles in the same segment theorem states that the angles formed by the same segment are equal. (angle a = angle b)
Summary of Segment of Circle
We can summarize the segment of a circle in the following key bullet points:
- Segment is the region formed by the chord and the arc.
- There are two types of segments: Major and minor.
- Semicircle is the largest segment of the circle.
- Area of the circle = Area of the major segment + Area of the minor segment.
- Area of segment = Area of the sector - Area of a triangle.
- Perimeter of the segment = Length of arc + Length of chord
Read More,
Solved Examples of Segment of a Circle
Problem 1: Find the area of the segment given that the area of the sector and the area of the triangle is 10 cm2 and 6 cm2.
Solution :
Area of the segment = Area of the sector - Area of the triangle
= 10 - 6
Area of the segment = 4 cm2
Problem 2: Find the perimeter of the segment if the length of the arc is 15 cm and the length of the chord is 10 cm.
Solution :
Perimeter of the segment = Length of the chord + Length of the arc
⇒ Perimeter of the segment = 15 + 10
⇒ Perimeter of the segment = 25 cm
Problem 3: Find the area of the segment if the radius of the circle is 15 cm and subtended angle is 30°.
Solution :
Area of segment (when θ in degrees) = (1/2) × r2 [(π/180) θ - sinθ]
⇒ Area of segment = (1/2) × 152 [(π/180) 30° - sin30°]
⇒ Area of segment = 2.65 cm2
Problem 4: Find the area of the circle if the area of the major and minor segments is 10 cm2 and 2 cm2.
Solution:
Area of the circle = Area of the major segment + Area of the minor segment
⇒ Area of the circle = 10 + 2
⇒ Area of the circle = 12 cm2
Problem 5: Find the perimeter of segment given the radius of the circle 25 cm and angle subtended by segment is 60°.
Solution:
Perimeter of segment (when θ in degrees) = rθ(π/180) - 2rsin(θ/2)
⇒ Perimeter of segment (when θ in degrees) = 25× 60(π/180) - 2× 25 × sin(60/2)
⇒ Perimeter of segment (when θ in degrees) = 25× (π/3) - 50 × sin(30)
⇒ Perimeter of segment (when θ in degrees) = 1.18 cm
Problem 6: Find the area of the segment if the height of the triangle is 10 cm and the angle subtended is 60°.
Solution:
We have to find radius of the circle to find the area of segment.
In triangle AOB, OP is the height (perpendicular) which bisects angle AOB.
∠ AOP = ∠ BOP = 30°
So, in right-angled triangle BOP
sin 30° = OP / OB
⇒ OB = 10 / sin 30°
⇒ OB (radius of circle) = 20 cm
Thus, Area of segment (when θ in degrees) = (1/2) × r2 [(π/180) θ - sinθ]
⇒ Area of segment (when θ in degrees) = (1/2) × 202 [(π/180) 60 - sin60]
⇒ Area of segment (when θ in degrees) = 109.44 cm2
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