Framing a box in a perspective camera is a per-corner solve, not a bounding sphere

Fact

The usual fit, distance = radius / sin(fov / 2) around a bounding sphere, is orientation-blind. For anything that is not roughly cubic it backs the camera off much further than needed, and the subject sits small in the middle of an empty frame — most visibly in a portrait window, where the horizontal field is the tight one and the sphere ignores which way the long axis points.

The exact answer is cheap. With the camera at distance d from the look-at point along away, a corner p measured from that point has depth p·forward + d, so it is inside the frustum when

d >= |p·right| / tanH - p·forward        (and the same with up / tanV)

Take the maximum over the eight corners. Recompute when the angle or the aspect changes; it is a few dozen operations.

Why it matters

Projecting the box's extents onto the camera axes — the obvious improvement over a sphere — is an orthographic approximation, and it clips: in perspective the near corners splay outward, so the corner closest to the camera leaves the frame while the arithmetic says it fits. The per-corner form has the depth term that accounts for exactly this.

How to apply

• Fit the box you actually want on screen, not the outermost trim. On a diagonal view only the corners reach the frame edge, so fitting a decorative border shrinks the subject for the sake of geometry nobody looks at.
• Pick the viewing angle from the aspect ratio: point a long subject away from the camera in a portrait frame and across it in a landscape one.
• Multiply by a small margin (a few percent) rather than a large one; the solve is exact, so the margin is only for what sticks out of the box.