Recently I came across the following formulas related to this:

Let's say "side" is the length of equilateral triangle's side.

Area of triangle = (sqrt(3) * square(side) ) /4.

Height of triangle = Height = (sqrt(3) * side ) / 2.

Radius of circle = ( 2 * Height ) / 3.

sqrt = square root.

Back to your question:

"side" here will be ( perimeter of triange /3 ).
Rest is a simple substitution of "side" in above formulas.

When you draw the picture, draw a line through the middle of the equilateral triangle to create two 30-60-90 triangles. Then, draw the 3 radii that go to the corners of the triangle. This will create two more, smaller 30-60-90 triangles.

Now, label the radii r. From here, use the side ratios of a 30-60-90, which are 1 : root(3) : 2, to fill in every side length in the picture. It's good practice, and it will show you all of the relationships mentioned by rppala90.

Thanks gmatboost.
Can you also please explain how radius of circle = (2 * Height) / 3.

Sure. Imagine (or draw) the equilateral triangle with one side along the bottom AB, and the other corner, C, at the top.

Now, let M be the midpoint of AB. The height is CM. Draw this line segment.
And let the center of the triangle/circle by O. So OA = OB = OC = r.

Now, OMB is a 30-60-90 triangle. Angle OBM is 30, because it is half of 60 degree angle ABC.
Similarly, angle BOM is 60, because it is half of 120 degree angle AOB.

In 30-60-90 triangle OMB, we know that OB is r. That means OM is r/2, since the side opposite the 30 degree angle is always half of the hypotenuse. That means the height CM is 1.5 r.

So, height = (3/2)r
Which is the same as r = (2/3) * height