Example 3
Find the area of the segment AYB shown in figure, if radius of the circle is 21 cm and ∠AOB = 120°. (Use π = 22/7 ).
In a given circle,
Radius (r) = 21 cm
And, 𝜃 = 120°
Area of segment AYB = Area of sector OAYB – Area of ΔOAB
Area of sector OAYB = 𝜃/360×𝜋𝑟2
= 120/360×22/7×(21)2
= 1/3×22/7×21×21
= 22 × 21
= 462 cm2
Finding area of Δ AOB
Area Δ AOB = 1/2 × Base × Height
We draw OM ⊥ AB
∴ ∠ OMB = ∠ OMA = 90°
In Δ OMA & Δ OMB
∠ OMA = ∠ OMB
OA = OB
OM = OM
∴ Δ OMA ≅ Δ OMB
⇒ ∠ AOM = ∠ BOM
∴ ∠ AOM = ∠ BOM = 1/2 ∠ BOA
Also, since Δ OMB ≅ Δ OMA
∴ BM = AM
⇒ BM = AM = 1/2 AB
From (1)
AM = 1/2AB
2AM = AB
AB = 2AM
Putting value of AM
AB = 2 × √3/2 × 21
AB = √3 × 21
AB = 21√3
Now,
Area of Δ AOB = 1/2 × Base × Height
= 1/2 × AB × OM
= 1/2 × 21√3 × 21/2
= (441√3)/4
Area of the segment AYB = Area of sector – area of ∆ 𝐴𝑂𝐵
= (462 – 441/4 √3 ) cm2

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.