Duct Transition (Reducer) Length from Slope
The length a duct size change needs to hold a target transition slope: concentric length = ((large - small)/2) / tan(slope), eccentric (one flat side) = (large - small) / tan(slope). SMACNA keeps the slope near 15 degrees per side (about a 4:1 run-to-offset ratio) to limit turbulence and pressure loss. A 20 in to 12 in concentric transition at 15 degrees is 14.93 in long; the same change eccentric (all on one side) needs 29.86 in; a steeper slope is shorter but noisier and higher-drop. On a rectangular duct the larger of the width and height changes sets the piece length. The SMACNA duct-construction standards and the system pressure loss govern.
Formula and source
concentric length = ((large_dim_in - small_dim_in) / 2) / tan(slope_deg); eccentric (one flat side) length = (large_dim_in - small_dim_in) / tan(slope_deg); run-to-offset ratio = 1 / tan(slope_deg).
Duct transition (reducer) length geometry, first-principles; SMACNA HVAC Duct Construction Standards keep the slope shallow to limit turbulence.
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