Name: Area of Annular Regions
Uploaded: 2019-07-19
Duration: 6 min 24 s
Description: The annular region of two given circles is the area enclosed between the concentric circles. In this video, we will discuss this with the help of a few examp...

Question

Find the area of the segment of a circle of radius $12 cm$ whose corresponding sector has a central angle of $60 ° .$ (Use $π = 3.14$ )

Accepted Answer

Given that, radius of circle $(r)$ $= 12 cm$

and central angle of sector $O B C A (θ) = 60 °$

$∴$ Area of sector $O B C A = \frac{π r^{2}}{360} \times θ$

[here, $O B C A =$ sector and $A B C A =$ segment]

$= \frac{3.14 \times 12 \times 12}{360 °} \times 60 °$

$= 3.14 \times 2 \times 12$

$= 3.14 \times 24 = 75.36 {cm}^{2}$

Since, $∆ O A B$ is an isosceles triangle

Let $∠ O A B = ∠ O B A = θ_{1}$

and $O A = O B = 12 cm$

$∠ A O B = θ = 60 °$

$∴ ∠ O A B + ∠ O B A + ∠ A O B = 180 °$ [ $∵$ sum of all interior angles of a triangle is $180 °$ ]

$θ_{1} + θ_{1} + 60 ° = 180 °$

$2 θ_{1} = 120 °$

$θ_{1} = 60 °$

$θ_{1} = θ = 60 °$

So, the required $∆ A O B$ is an equilateral triangle,

Now, area of $∆ A O B = \frac{\sqrt{3}}{4} {(side)}^{2}$ [ $∵$ area of an equilateral triangle $= \frac{\sqrt{3}}{4} {(side)}^{2}$ ]

$= \frac{\sqrt{3}}{4} {(12)}^{2}$

$= \frac{\sqrt{3}}{4} \times 12 \times 12 = 36 \sqrt{3} {cm}^{2}$

Now, area of the segment of a circle ie.,

$A B C A =$ Area of sector $O B C A$ - Area of $∆ A O B$

$= (75.36 - 36 \sqrt{3}) {cm}^{2}$

Hence, the required area of segment of a circle is $(75.36 - 36 \sqrt{3}) {cm}^{2} .$

Find the area of the segment of a circle of radius $12 cm$ whose corresponding sector has a central angle of $60 ° .$ (Use $π = 3.14$ )

Important Questions on Areas Related to Circles

Learn and Prepare for any exam you want !

Find the area of the segment of a circle of radius 12 cm whose corresponding sector has a central angle of 60°.(Use π=3.14)

Important Questions on Areas Related to Circles

Learn and Prepare for any exam you want !

Find the area of the segment of a circle of radius $12 cm$ whose corresponding sector has a central angle of $60 ° .$ (Use $π = 3.14$ )