Give a Plane Volume via Shader

I have a planet with rings that I am working on rendering. I have mostly finished the planet shader and have been fixing up the ring shader. The rings look good when viewed from above. The issue occurs, like with most planes, when viewed parallel to the plane. This results in the rings disappearing in the worst case or becoming barely a pixel thin. I'm looking for a means to add volume to this ring plane. Ideally I'd like to keep it all in the shader instead of moving to using something like a cube.

From what I've read, it looks like you can use a vertex shader to move the position of vertices or extend them, however I'm not familiar with the details of this and haven't found anything doing something similar yet.

Right, so my question is this: how can I give my plane a volume so it appears normal when viewed from the side? Instead of looking like... well... this.