Let the height of the cone be h and of the sector it is formed from be R.

Then the radius of the base of the cone is

r = sqrt(R^2-h^2)

and so the volume of the cone is:

V = (1/3) h pi r^2 = (pi/3) h (R^2-h^2)

This volume is a maximum when dV/dh = 0, but:

dV/dh = (pi/3) (R^2 - 3 h^2),

which is zero when h = R/3, which is clearly a maximum, so the maximum

volume V = (8 pi/27) R^3.

RonL