The largest triangle that can be fit in a circle of radius 12 is one that is circumscribed by the circle. As the area of the triangle has to be be maximized it is an equilateral triangle.

The area of an equilateral triangle if the radius of the circumscribed circle is 12 has to be determined.

The area of an equilateral triangle with sides a is `(sqrt 3/4)*a^2` . The radius of the circumscribed circle is `a/sqrt 3`

`a/sqrt 3 = 12`

=> `a = 12*sqrt 3`

The area of the triangle is `(sqrt 3/4)*a^2 = (sqrt 3/4)*(12*sqrt 3)^2`

=> `(sqrt 3/4)*(144*3)`

=> `108*sqrt 3`

**The required area of the largest triangle that can fit in a circle of radius 12 is `108*sqrt 3` **

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