Roof Pitch: Converting a Japanese Sun Pitch to Degrees

From the Japanese sun pitch (the rise for a run of 10), finds the angle and the gradient as a percentage. A 4-sun pitch is about 21.8 degrees, or 40%.

Explanation

Japanese roofs are specified in sun: a 4-sun pitch, a 5-sun pitch. The sun is an old unit of length, about 3.03 cm, a tenth of a shaku. In a roof pitch, though, it is really a ratio: the rise, in sun, for every 10 sun (one shaku) of horizontal run. The units cancel, so a 4-sun pitch simply means 4 up for 10 along. It is the same idea as the American 4-in-12, but with a base of 10 rather than 12.

\theta = \arctan\left(\dfrac{s}{10}\right)

\text{gradient}\ (\%) = \dfrac{s}{10} \times 100 = 10s

$s$ is the pitch in sun and $\theta$ is the angle from horizontal.

Example

Take the default, 4 sun.

\theta = \arctan(0.4) = 21.80\ldots\ \text{degrees}

\text{gradient} = 0.4 \times 100 = 40\ \%

A 4-sun pitch is about 21.8 degrees, or 40%. It is the workhorse pitch of Japanese timber houses. A 10-sun pitch, known as kane-kobai, is exactly 45 degrees.

Where you use it

A sloping roof has more surface than its footprint. The multiplier is $\sqrt{1 + 0.4^2} = 1.077$ for a 4-sun pitch, so take the plan area off the drawing and multiply by it before ordering roofing.
Percent and degrees are not the same thing. 100% is 45 degrees, not vertical. A drawing that calls for a 40% slope is not asking for 40 degrees.
Roofing manufacturers publish a minimum pitch for each product. Too shallow and water works its way under the laps; too steep and you pay for scaffolding and slower work. Settle the material first, then fix the pitch.