Why do we use the focal length to describe a lens and not the viewing angle?

Because the viewing angle can change!

The focal length is a feature of a lens that doesn’t change when one changes the camera.

However, with a change of sensor size the viewing angle changes.

Also when we focus lenses to a different distance, usually the viewing angle changes!

When we use  (with the same lens!) a camera with a smaller sensor, the viewing angle also gets smaller.

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