How to find the area of a segment of a circle?

A segment of a circle is the region enclosed by a chord and an arc so formed touching the endpoints of the chord.

Answer: The area of a segment of a circle is obtained by subtracting the area of the triangle at the center, from the area of the sector of the circle. Let's look at how to find the area of a segment of a circle with the help of the example below.

Read the steps carefully and construct.

Explanation:

Observe the sector of a circle in the below diagram. It consists of a two-part one part is a triangle, and another part is a segment. The area of the sector is equal to the area of the triangle plus the area of the segment.

So, the area of the segment is found by subtracting the area of the triangle from the area of the sector.

Let's look at an example.

The radius of the circle is 7 cm and Central angle θ is 120°

Area of segment APB = Area of sector OAPB - Area of ∆OAB

Area of sector =  π r² × θ / 360⁰

= 22/7 × 7² × 120° / 360°

= 51.3 cm²

Area of sector OAPB = 51.3 cm²

You can use Cuemath's Area of a Sector Calculator to verify your answer.

To find the area of AOB, draw OM 丄 AB.

In ∆OMA and ∆OMB,

OM = OM

∠AMO = ∠BMO = 90^°

AM = BM

So, ∆OMA  ≅ ∆OMB

∠AOM = ∠BOM = 60°

cos ∠AOM = OM/OA

OM = cos ∠AOM × OA = 3.5 cm

Since sin ∠AOM = AM/OA

AM = sin ∠AOM × OA = √3 / 2 × 7

AM = √3 × 7 / 2 

AB = 2 × AM

AB = 7 √3

Area of ∆OAB = 1/2 × AB × OM = 21.2 cm² 

Now, area of segment APB = Area of sector OAPB - Area of ∆OAB

= (51.3 - 21.2) cm²

= 30.1 cm²

Area of segment APB = 30.1 cm²

Thus, the area of the segment is found by subtracting the area of the triangle from the area of the sector.