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The roof's pitch is its vertical rise divided by its horizontal span (or "run"), what is called "slope" in geometry and stair construction, or the tangent function in trigonometry. In imperial measurement systems it is typically expressed with the rise first and run second, with the run denominated by the number 12, giving a ratio of how many inches of incline there is to each foot of run. For example, 3:12, 4:12, 5:12, and so on. Countries which use metric measurement systems use a degree angle.
The pitch matters for a variety of reasons including the type of roofing material used, walkability, proportions to the building as a whole which is sometimes a critical factor in some architectural styles such as a steep pitch in Gothic architecture and a low pitch in Classical architecture, and combinations of pitches form distinctive roof shapes such as a gambrel roof. The basic ranges of pitch are not uniformly defined but range from flat, which are not perfectly flat but sloped to drain water up to 1/2:12 to 2:12; low-slope roofing requires special materials and techniques to avoid leaks and ranges from 1:12 (2:12) to 4:12; conventional from 4:12 to 9:12; and steep-slope roofing is above 9:12 (21:12) and may require extra fasteners.
Convention is to use whole numbers when even (e.g. "three in twelve") or the nearest single or two-digit fraction when not (e.g. either "five and a quarter in twelve" or "five point two-five in twelve", each expressed numerically as 5.25:12).
To mathematically find the exact roof slope in degrees, one takes the arctangent. For example: arctan(3/12)=14,0°
The primary purpose of pitching a roof is to redirect water and snow. Thus, pitch is typically greater in areas of high rain or snowfall. The steep roof of the tropical Papua New Guinea longhouse, for example, sweeps almost to the ground. The high, steeply-pitched gabled roofs of northern Europe are typical in regions of heavy snowfall. In some areas building codes require a minimum slope. Buffalo, New York and Montreal, Quebec, Canada, specify 6 in 12, a pitch of approximately 26.6 degrees.
Carpenters frame rafters on an angle to "pitch" a roof. Gable and other multi-pitched roofs allow for lower primary structures with a corresponding conservation of exterior framing and sheathing materials.
Commonly used roof pitches were given names such as using the rafter-length method: Common, the rafter length is 3/4 the span; Gothic, the rafters equal the span; Greek, 1/9 to 1/7th the span; Roman, 1/3 to 2/9ths the span giving an angle of 23° to 24°; and Elizabethan, the rafters are longer than the span.
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