Ceiling Speaker Coverage Angle for a Target Spacing
The inverse of the ceiling-speaker-coverage tile: the coverage angle a target coverage diameter (or on-center spacing) needs at a mounting drop, angle = 2 x atan( diameter / (2 x (ceiling - ear)) ). An 8 ft circle under a 10 ft ceiling over 4 ft ears needs a 67 degree speaker. Spec a speaker rated at least this wide (the angle narrows at high frequency). Answers 'what coverage angle to spec' instead of the diameter from a set angle. A layout aid; verify with the speaker spec and target SPL.
Formula and source
coverage_deg = 2 x atan( target_diameter / (2 x (ceiling - ear)) ), the inverse of diameter = 2 x (ceiling - ear) x tan(coverage_deg / 2).
Distributed ceiling-loudspeaker coverage geometry, standard commercial-audio design practice (manufacturer design guides, e.g. Bose / JBL / Atlas) by name, solved for the angle; first-principles cone geometry.
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