A picture is worth 1000 words, so here's 1000 words for you...

The text of the original question would be helpful here.

If I am visualizing correctly, you have the equation of a plane with normal vector N, the location of the apex of the cone, A, and a point on the circular edge of the cone, B, its circular area contained in the plane.

If this is correct, you can find the equation of the line in the direction of N passing through A, and the point at which this line passes through the plane will be the center of its circular area, C. Therefore, the radius of the cone's circular area will be the distance BC, and the height of the cone will be AC.