The height of the cone is (1-((360-x)/360)^{2})^{1/2}.

The area of the cone is 1/3*pi*radius^{2}*height.

Let u=((360-x)/360)^{2}.

The area becomes: pi/3 * (1-u)^{1/2} * u.

Take the deriviate and set equal to 0: u=2/3.

If u=2/3 then x=360*(1-(2/3)^{1/2}).

Thus the radius = (2/3)^{1/2}.

Thus the height = 3^{-1/2}.

The area of the cone is (2*pi)/(9*3^{1/2}) =~ .403.

Thanks for "Roeterink" for correcting me first answer.

Michael Shackleford, A.S.A.