What went wrong while computing the area of this surface?

Question

I have this surface: $\{x^2+y^2+z=5, z\ge 4\}$, and I want to know its area.
By writing the first equation like this: $x^2+y^2=5-z$, I thought I could compute the perimeter of the disc on the plane xy of ray $\sqrt{5-z}$ first and then calculate the integral of said perimeter respect to z, like this: $\int_4^5 2\pi\sqrt{5-z}~dz$, so my answer is $\frac{4\pi}{3}$.
Now, when I saw the solutions I saw that it is $\frac{5\sqrt{5}\pi}{6}$ (or something like that, I don't have them now). What did I do wrong?