Best room shape for 2,895 ft³–5,000 ft³
One shape serves every room between 2,895 ft³ and 5,000 ft³: length 3.200 and width 1.231 times the ceiling height. It is the exact optimum at every volume in the band.
Shape (L : W : H)
3.200 : 1.231 : 1
scale-free — multiply by the ceiling height you have
Covers
2,895 ft³–5,000 ft³
2,106 cubic-foot values
Worst case in band
optimal
zero to machine precision across the whole range
At 3,948 ft³
12.32 × 32.03 × 10.01 ft
widest axial gap 17.57 Hz, 18 axial modes below 200 Hz
Dimensions across the band
The same proportions at a spread of volumes inside the band. Only the absolute sizes change — they scale as the cube root of the volume — so find the row nearest the space you have and build to it. The last column is what that room gives up against the best shape known at its own exact volume.
| Volume | Width × Length × Height | Widest axial gap | vs optimal |
|---|---|---|---|
| 2,895 ft³ | 11.11 × 28.88 × 9.02 ft | 19.48 Hz | optimal |
| 3,196 ft³ | 11.48 × 29.85 × 9.33 ft | 18.85 Hz | optimal |
| 3,496 ft³ | 11.83 × 30.75 × 9.61 ft | 18.30 Hz | optimal |
| 3,797 ft³ | 12.16 × 31.61 × 9.88 ft | 17.80 Hz | optimal |
| 4,098 ft³ | 12.47 × 32.43 × 10.13 ft | 17.35 Hz | optimal |
| 4,399 ft³ | 12.77 × 33.20 × 10.38 ft | 16.95 Hz | optimal |
| 4,699 ft³ | 13.05 × 33.94 × 10.61 ft | 16.58 Hz | optimal |
| 5,000 ft³ | 13.33 × 34.65 × 10.83 ft | 16.24 Hz | optimal |
Why one shape covers the whole range
The objective is the widest gap between adjacent axial modes below 200 Hz. Under pure scaling every axial frequency moves as the cube root of the volume, so the best proportions do not change with volume at all — until a mode crosses the 200 Hz window edge and the maximum jumps to a different pair. This band sits entirely inside one such regime, which is why its worst case is not merely small but zero: the published shape is the best shape known at every volume in the range.
The shape sits on the Bolt boundary (L/H<=3.2). Above roughly 3,000 ft³ the pull toward elongation reasserts itself and the criterion is what stops it, so this is a reported state rather than a failure.
Questions this band answers
What is the best room shape for 2,895 ft³–5,000 ft³?
A ratio of 3.200 : 1.231 : 1 — length : width : height. At the middle of the band, 3,948 ft³, that is 12.32 × 32.03 × 10.01 ft, and the widest gap between adjacent axial modes below 200 Hz is 17.57 Hz. The same ratio applies at every volume in the band; only the absolute dimensions scale, as the cube root of the volume.
How much worse than optimal is this shape for my exact volume?
Not at all. Across the whole 2,895 ft³–5,000 ft³ range this shape is the best shape known at every volume in it — the worst case measured over the band is zero to machine precision. The reason is that the objective is a maximum over axial modes inside a fixed 200 Hz window: under pure scaling every axial frequency moves as the cube root of volume, so the optimal ratio does not change until a mode crosses the window edge. This band sits inside one such regime.
Why is this a range rather than a page per volume?
Because a recommended room gets built to a finite tolerance. A ±1 inch error in framing moves this objective by a median 6.86%, so splitting the range more finely than that publishes distinctions nobody can construct. Three shapes cover 800–5,000 ft³ at that tolerance, and the covering is provably minimal there rather than merely an upper bound, because each shape's coverage set is a single unbroken interval.
Is this shape inside the Bolt area?
Yes — every shape published here is. The shape sits ON the Bolt boundary (L/H<=3.2). Above roughly 3,000 ft³ the elongation pull reasserts itself and the criterion is what stops it, so this is a reported state rather than a failure. Bolt (1946) is a citable constraint set rather than a sweep range someone typed: it caps length at 3.2 times the height, which puts the corridor outside the feasible set, and forces height ≤ width < length. Without it the answer to "what shape?" at a fixed volume is "build a longer corridor", which is a tautology rather than advice.
What does the widest axial gap actually tell me?
It is the largest hole in the low-frequency response you cannot fix with absorption. Axial modes are the strongest room resonances, and a wide gap between two adjacent ones is a band where the room supports very little. Minimising the widest gap spreads the modes as evenly as the volume allows. It says nothing about tangential or oblique modes, absorption, speaker placement or anything above 200 Hz.
The neighbouring bands
A different range, and a genuinely different shape — a boundary is where this shape stops being within tolerance, not an arbitrary cut.
Method and limits. Solved by minimising the widest gap between adjacent axial modes below 200 Hz, at fixed volume, over the Bolt (1946) ratio area. Axial modes are n·c/(2L) per axis with c = 343 m/s. The band is a covering: one shape is published for a range of volumes, with the range chosen so that no volume inside it pays more than 7% against the best shape known at that volume — the first tolerance above the 6.86% a ±1 inch framing error costs. "Best known" is the minimum over a live solve and every shape the solver returns anywhere on the range, which matters because the search is itself imperfect: it is beaten at 2.43% of volumes by a shape it finds elsewhere. Worst-case figures are measured across the band, not assumed. Tangential and oblique modes, absorption and placement are out of scope. Pure computation — nothing is fetched and nothing is recalled.