In a regular triangular pyramid, the radius of the circle inscribed in the base is 3

In a regular triangular pyramid, the radius of the circle inscribed in the base is 3, the height of the pyramid is 4. Find the volume of the described sphere.

It is known that:
AE is the diameter of a circle circumscribed around the base of the pyramid;
T is the diameter of the ball;
AE and MT are chords, and they are equal;
AO = OE = 6;
Let’s make the equation:
we will assume that TP is the radius, then the segment OT = x.
Then, we get an equation of the following form:
MO + OT = 4 + x ⇒ MO * OP = AO * OE;
Let’s transform:
4 * (4 + x) = 6 * 6;
Let’s count:
16 + 4 * x = 36;
4 * x = 20;
Find x:
x = 2.5;
Find the radius:
R = 4 + 2.5;
R = 6.5;
Find V:
V = (4/3) * 3.14 * (6.5) ³;
V = 1149.76;
Answer: V = 1149.76;