kook44 wrote:

Is there any way to solve these without knowingn these?

just using the standard properties of triangles & circles?

Yes, I used another approach

Assume the triangle is ABC and O is the center of the circle.

AO, BO and CO divide the triangle ABC into 3 equal isosceles triangles,

when each has an area of 9sqrt(3) / 3 = 3sqrt(3) .

Extending the AO up to BC , we get a height AD , so OD is the height of triangle BOC.

Now assume OD = x , hence DC = x*sqrt(3) and OC=2 *x = R ( ODC is a right triangle 30-60-90 )

the area of ODC = 3sqrt(3) / 2

x*x*sqrt(3) / 2 = 3sqrt(3)/2

x = sqrt(3)

OC = R = 2*sqrt(3) , hence the area of the circle is 12pi