The vertices of the triangle divide the circle into three arcs of length 3, 4 and 5 units.

So the circumference of the circle is

3+4+5=2\pi R

Then, R = \dfrac{6}{\pi}

The arcs subtend angles that are proportional to the length of arcs and sum to a round angle, that is

3θ+4θ+5θ = 360\degree

Then, θ =30\degree

The central angles subtended by arcs are 90,\degree ,120\degree and 150\degree

The area of the triangle is the sum of the three triangles formed by radii of the circle and the sides of the triangle.

Using the area formula \dfrac{1}{2}ab\sin C , here the sides are the radii of the circle which are the same for the three triangles. The angles are the central angles subtended by arcs. Then,

A = \dfrac{1}{2} R^2\sin 90\degree + \dfrac{1}{2} R^2\sin 120\degree + \dfrac{1}{2} R^2\sin 150\degree

= \dfrac{1}{2} \cdot \dfrac{36}{\pi^2}(1+\dfrac{\sqrt{3} }{2}+\dfrac{1}{2} )

=\dfrac{9(3+\sqrt{3}) }{\pi^2}

Therefore, A is the correct choice.