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

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.