Question

An equilateral triangle is inscribed in a circle. Select the option that indicates
how many times is the area of circle with respect to the area of triangle.
​

Answer 1

Answer:

⇒ Here, AB=BC=AC=12cm

⇒ Let OP=OR=OQ=r

⇒ We have O as the incenter and OP,OQ and OR are equal.

⇒ ar(△ABC)=ar(△OAB)+ar(△OBC)+ar(△OCA)

4

3

×(side)

2

=(

2

1

×OP×AB)+(

2

1

×OQ×BC)+(

2

1

×OR×AC)

⇒

4

3

×(12)

2

=(

2

1

×r×12)+(

2

1

×r×12)+(

2

1

×r×12)

⇒

4

3

×(12)

2

=3(

2

1

×12×r)

∴ r=

18

36

3

∴ r=2

3

cm

⇒ Area of the shaded region = Area of △ABC - Area of circle.

⇒ Area of the shaded region =

4

3

×(12)

2

−

7

22

×(2

3

)

2

⇒ Area of the shaded region =(62.35−37.71)cm

2

=24.64cm

2