Question

the circumference of a circle circumscribing an equilateral trianlge is 24*22/7 units. Find the area of a circle inscribed in the equilateral triangle.

Answer 1

I have solved this problem in the
attachment.

Radius of the given circle is found from
the circumference.

Make use of the following result I have found the side of an equilateral triangle ABC:

The side of an equilateral triangle which can be drawn on a circle of radius r is
r Root(3) .

The radius of circle inscribed in the
Triangle is calculated by applying
Pythagoras theorem in right triangle DOC.

Hence radius is 6 units.
.
.
Karups

Answer 2

Hello Dear.

Here is the answer---

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Refers to the attachment for the Question,

 Circumference of the Circle = 24 × (22/7)
     2 × π × r = 24 × (22/7)
     2r × (22/7) = 24 × (22/7)
      ⇒ 2r = 24
     ⇒ r = 12 cm.

We know, the Side of the Equilateral Triangles inscribed in a circle of radius r = √3 × r
= √3 × 12
= 12√3 cm.


Now, From the Attachment,
We know, OA is Perpendicular on the QR.

[∵ The line drawn from the center to the chord is always Perpendicular]

From the Figure, we can see that OA is the Radius of the Circle.
AR = (1/2)QR 
[∵ The Perpendicular Drawn from the Centre bisects the Chords]
⇒ AR = 6√3 cm.


In Δ OAR,
Applying Pythagoras theorem,

   OR² = OA² + AR²
⇒ (12)² = OA² + (6√3)²
⇒ 144 = OA² + 108
⇒ OA² = 144 - 108
⇒ OA² = 36
⇒ OA = 6 cm.


∴ Area of the Circle = π × r²
   = (22/7) × (6)²
   = 792/7 cm²


∴ the Area of the In circle is 792/7 cm². 


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Hope it helps.