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The area of a circle inscribed in a equlateral triangle is 154 cm2. Find the perimeter of the triangle?
Ref: https://edurev.in/question/480447/The-area-of-a-circle-inscribed-in-a-equlateral-triangle-is-154-cm2-Find-the-perimeter-of-the-triang

Solution - Class 10
Given area of inscribed circle = 154 sq cmLet the radius of the incircle be r.⇒ Area of this circle = πr2 = 154
(22/7) × r2 = 154
⇒ r2 = 154 × (7/22) = 49
∴ r = 7 cm
Recall that incentre of a circle is the point of intersection of the angular bisectors.
Given ABC is an equilateral triangle and AD = h be the altitude.
Hence these bisectors are also the altitudes and medians whose point of intersection divides the medians in the ratio 2 : 1
∠ADB = 90° and OD = (1/3) AD
That is r = (h/3)
Þ h = 3r = 3 × 7 = 21 cm
Let each side of the triangle be a, then
Altitude of an equilateral triangle is (√3/2) times its side
That is h = (√3a/2)
Solution - Class 10
∴ a = 14√3 cmWe know that perimeter of an equilateral triangle = 3a
= 3 × 14 √3 = 42√3
= 42 × 1.73 = 72.66 cm

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