Shaku to Tsubo Derivation
Overview
The apparently arbitrary 3.3058 that converts tsubo and pyeong to square metres is the endpoint of a short exact derivation. Meiji-era legislation defined one shaku as 10/33 of a metre, a ratio chosen to align the traditional measure with the metric standard Japan adopted in 1886. One ken, the structural bay spacing of traditional carpentry, is six shaku, or 60/33 = 20/11 metre — approximately 1.818 m. A tsubo is one square ken, so its area is (20/11)² = 400/121 square metres, or 3.30578512… The value is rational and exact; every decimal representation is a truncation. The same 20/11 metre module governs tatami proportions and the column grid of wooden buildings, which is why traditional room sizes fall on tidy mat counts and why floor plans convert to whole tsubo without remainder.
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $shaku$ | Base length unit | 尺 | Defined by Meiji statute as exactly 10/33 metre. |
| $ken$ | Structural bay | 間 | Six shaku, that is 20/11 metre ≈ 1.8182 m. |
| $tsubo$ | Square ken | 坪 | One square ken, 400/121 m². |
Where 3.3058 Comes From
The conversion constant for tsubo and pyeong looks arbitrary but follows from a three-step exact derivation.
$$1\ shaku = \frac{10}{33}\,m \quad\Rightarrow\quad 1\ ken = 6 \times \frac{10}{33} = \frac{20}{11}\,m \quad\Rightarrow\quad 1\ tsubo = \left(\frac{20}{11}\right)^2 = \frac{400}{121}\,m^2$$
The result, 3.30578512396…, is a repeating rational number. Every decimal form in common use — 3.3058, 3.306, 3.31 — is a truncation of it.
Why 10/33
Japan joined the Metre Convention in 1885 and needed to express the existing shaku in metric terms without disturbing established construction practice. The traditional shaku measured close to 30.3 cm, and 10/33 metre is 30.303 cm — near enough to leave carpentry unchanged while giving an exact legal definition. The choice of a simple fraction over a decimal approximation kept derived units rational, which is why the tsubo has an exact value rather than a rounded one.
The Module in Building
The ken is not merely six shaku on paper; it is the column spacing of traditional timber framing. Rooms are laid out on a ken grid, tatami mats are half-ken by one ken, and sliding-door openings follow the same module. A building's footprint measured in tsubo therefore counts structural bays directly, which is why floor plans resolve to whole tsubo without remainder and why the unit persists among builders long after its legal retirement.
The Wider Shakkanhō System
| Unit | Definition | Metric |
|---|---|---|
| sun (寸) | 1/10 shaku | 3.03 cm |
| shaku (尺) | base | 30.303 cm |
| ken (間) | 6 shaku | 1.8182 m |
| jō (丈) | 10 shaku | 3.0303 m |
| chō (町) | 60 ken | 109.09 m |
Derivation & History
The Meiji Weights and Measures Act of 1891 fixed the shaku at 10/33 metre and the kan, the mass unit, at 15/4 kilogram, both chosen as simple fractions close to the existing prototypes. The 1951 Measurement Act made the metric system mandatory for transactions and phased out shakkanhō for official use, completing the transition in 1966. Korea's own Measurement Act followed the same pattern, retiring the pyeong from commercial use in 2007. In both cases the legal definition survived its own prohibition, because the derived units remain in daily speech.
Worked Examples
Deriving the constant from scratch
- One ken = 6 shaku = 60/33 = 20/11 m ≈ 1.818182 m
- One tsubo = (20/11)² = 400/121 m²
- 400 ÷ 121 = 3.30578512…
Result: 3.3058 m² per tsubo (exact value 400/121)
A 4-ken by 3-ken house footprint
- Footprint in tsubo: 4 × 3 = 12 tsubo
- Metric: 12 × 400 ÷ 121 = 39.67 m²
- Or directly: (4 × 20/11) × (3 × 20/11) = 7.273 × 5.455 = 39.67 m²
Result: 12 tsubo, 39.67 m²
Edge Cases & Limitations
Pre-Meiji shaku: Regional variants such as the kanejaku and the kujirajaku differed by several percent, so historical documents cannot use the modern factor.
Kujirajaku: The textile shaku, 25/66 metre, is 1.25 times the carpenter's shaku and applies to cloth measurement rather than building.
Truncation depth: Using 3.31 instead of 400/121 overstates a 100-tsubo plot by about 0.42 m².
Korean adoption date: Standardisation reached Korea through colonial-era legislation, so pre-1910 Korean land records may use different local measures.
Real-World Applications
Traditional carpentry and temple restoration work in shaku and ken throughout. Japanese building products — plywood, tatami, sliding screens, timber lengths — are manufactured to the module, so renovation of older buildings requires the units. Property valuation converts between tsubo and square metres continuously.