Minimum distance between a disk in 3d space and a point above the disk How can I calculate the minimum distance between a point on the perimeter of a disk in 3d space and a point above the disk?
For example, there is a disk in 3d space with center [0,0,0]. It has radius 3 and lies flat on the x,y plane. If there is a particle above the disk at [5,5,5], how do I calculate the minimum distance from this particle to a point on the perimeter of the disk?

Here is my attempt so far:
vec1 = vector between disk center and particle
vec1 = [(5 - 0), (5 - 0), (5 - 0)]
vec1 = [5,5,5]

unitvec1 = unit vector in direction of vec1
unitvec1 = vec1/norm(vec1)
unitvec1 = [0.5774, 0.5774, 0.5774]

vec2 = vector between disk center and point on the perimeter closest to the particle
vec2 = disk radius * unitvec1, and make z element = 0
vec2 = 3 * [0.5774, 0.5774, 0]
vec2 = [1.7321, 1.7321, 0]

vec3 = vector between particle and point on the perimeter closest to the particle
vec3 = vec1 - vec2
vec3 = [3.2679, 3.2679, 5.0000]

So the min distance is
norm(vec3) = 6.8087

But this method doesn't always work. If I try it with disk center [0,0,0], particle location [0,0,6], and disk radius 9, it gives the minimum distance to be 6. this can't be correct, because the distance between the center of the disk and the particle will be 6, so the distance to the perimeter must be larger.
What am I doing wrong, and how should I actually calculate this?
Thanks!
note: I am using pseudo code, not an actual programing language 

I'm going to answer my own question, I've just been told a much easier way to calculate this.
Let R = radius of disk
Let r = distance from disk center to projection of particle on x,y plane
so r = ((5^2)+(5^2))^0.5 = 4.071
Distance between concentric circles = ((R-r)^2)^0.5 = 4.071
Height of particle above disk = 5
Rmin can then be calculated using Pythagorean theorem 
Rmin = ( (4.071^2) + (5^2) )^0.5 = 6.448
