The lateral area of the cone is pi*r*L
The base area of the cone is pi*r^2
The surface area of the insphere is 4*pi*a^2
Then the ratio Ac/As = (pi*r*L+pi*r^2)/(4*pi*a^2) = r(L+r)/(4*a^2)
Take a cross section of the cone-sphere containing the altitude. That creates an isosceles triangle and its incircle. The triangle has sides L, L, and 2r and the circle has radius a.
The semiperimeter of the triangle is s=r+L. Then the inradius a = sqrt[(s-L)*(s-L)*(s-r)/s] = r*sqrt[(L-r)/(L+r)]
Then Ac/As = r(L+r)/(4*r^2*(L-r)/(L+r)) = (L+r)^2/(4*r*(L-r))
Let x=L/r. Then Ac/As = r^2*(x+1)^2/(4*r^2*(x-1)) = (x+1)^2/(4*(x-1))
The derivative d(Ac/As)/dx = (x+1)*(x-3)/(4*(x-1)^2). It has a positive root of x=3, which is the local minimum.
With x=3 then Ac/As=2. Then a : r : L : h = 1 : sqrt(2) : 3*sqrt(2) : 4
Edited on December 25, 2016, 1:37 pm