Hey guys, I'm having a problem with some basic math for a basic concept.

Idea:

I have a rectanglar box in my scene and I want to distance the camera such that the box just fits.

Background:

So right now I have my camera set up so that I can select a heading and an elevation. The software then calculates the camera's position using that data as if the camera is connected to a piece of string attached to the middle of the cube, tethering it to a spherical area. EG: I give it a heading of 45 degress and an elevation of 30 degrees, the camera is positioned 'Southwest' of the cube so it's looking 'Northeast' (45 degrees) and has an elevation of 30 degrees from the ground. If I give it a heading of 180 and an elevation of 90, the top of the screen is 'South' and the camera is directly above the box. (The x-axis is East/West, the y-axis is North/South). Now I'm trying to figure out to calculate the distance the camera needs to be so the box fills the screen. I know how to calculate the distance the camera needs to be so that no matter how the cube is rotated it fits (distance = boundRadius / sin(FOV / 2)). The problem with this is that the box is a wide, narrow rectangle, with little depth so using this method the box fills the screen only at a specific heading and elevation, otherwise there is a lot of empty space.

Current idea:

Since, ultimately, there's no difference between positioning the camera and rotating the mesh, I can think of the camera as being stationary and the mesh is being rotated so it's displayed with the correct heading and elevation. I was thinking that if I calculate the width and height of the box after it's rotated I can use trigonometry to calculate the distance (distance = halfWidth / tan(FOV / 2)). (If there's an easier/better idea, by all means, please let me know). I know how to calculate a 2D square's height and width after a 1D rotation, but I don't know how to calculate a 3D box's height and width after a 2D rotation.

What I know:

The box's height, width and depth; the camera's field of view, elevation (IE: x-axis rotation) and heading (IE: z-axis rotation).