Best room shape, by volume

3 shapes cover every volume from 800 to 5,000 cubic feet. Each one minimises the widest gap between adjacent axial modes below 200 Hz, inside the Bolt area — and 1 of the 3 is the best shape known at every volume it serves.

Solve an exact volume →Analyse a room you have

800–1,332 ft³

1.664 : 1.332 : 1

length : width : height

worst case 1.66%

at 1,066 ft³: 10.4 × 13.0 × 7.8 ft

1,333–2,894 ft³

2.247 : 1.281 : 1

length : width : height

worst case 4.72%

at 2,114 ft³: 11.6 × 20.3 × 9.0 ft

2,895–5,000 ft³

3.200 : 1.231 : 1

length : width : height

optimal across the band

at 3,948 ft³: 12.3 × 32.0 × 10.0 ft

Why 3 pages and not five hundred

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 a reader cannot construct. At a tolerance of 7% the whole 800–5,000 ft³ range is covered by 3 shapes, and that covering is provably minimal rather than an upper bound — each shape's coverage set is a single unbroken interval. The page count is derived from the measurement, not chosen.

What each band guarantees

Band	Shape (L : W : H)	Guarantee	Bolt
800–1,332 ft³	1.664 : 1.332 : 1	no room in the band is worse off than 1.66%, and that worst case holds across 800 ft³–1,273 ft³	interior
1,333–2,894 ft³	2.247 : 1.281 : 1	no room in the band is worse off than 4.72%, and that worst case holds across 1,333 ft³–2,550 ft³	interior
2,895–5,000 ft³	3.200 : 1.231 : 1	this shape is the best known at every volume in the band	L/H<=3.2

Method and limits. Each band's shape minimises the widest gap between adjacent axial modes below 200 Hz at fixed volume, over the Bolt (1946) ratio area, solved by a compass search seeded on a feasible grid. Axial modes are n·c/(2L) per axis with c = 343 m/s. The bands are a covering: one shape per range of volumes, with the ranges chosen so no volume inside a band pays more than 7% against the best shape known at that volume. Tangential and oblique modes, absorption, furniture and speaker placement are all out of scope. Pure computation — nothing is fetched and nothing is recalled.