Roof Area from the Pitch

Finds the true area of a roof from its footprint seen from above and its pitch. A sloped surface is longer than its horizontal projection, so the footprint is multiplied by a pitch factor. This is the figure needed when ordering roofing.

A roof is always larger than it looks from above. Being tilted, its surface is longer than its horizontal projection.

The area seen from directly above is the footprint. Multiplying it by a pitch factor gives the true roof area.

k = \sqrt{1 + \left(\dfrac{p}{10}\right)^2} \qquad S = A k

$k$ is the pitch factor
$p$ is the pitch in sun, the rise for a run of 10
$A$ is the footprint and $S$ the roof area

Inside the factor is the Pythagorean theorem. Rising $p$ over a run of 10 makes the sloping length $\sqrt{10^2 + p^2}$ , and dividing by 10 gives the slope length per unit of horizontal distance.

Example

The default input is a footprint of 60 square metres at a 4-sun pitch.

The pitch factor is $\sqrt{1 + 0.16}$ , about 1.0770, so the roof area is $60 \times 1.0770$ , about 64.62 square metres. As an angle that is about 21.8 degrees.

With a ten percent allowance, about 71.08 square metres of roofing is needed.

Common pitch factors

For the pitches most often used: 3 sun gives about 1.0440, 4 sun about 1.0770, 5 sun about 1.1180 and 6 sun about 1.1662. A 10-sun pitch is exactly 45 degrees, with a factor of $\sqrt{2}$ , about 1.4142.

Japanese housing settles mostly between 4 and 5 sun. That sheds rain and snow well while remaining shallow enough to work on from staging.

Points to watch

Ordering roofing by the footprint leaves you short — by eight percent at a 4-sun pitch and seventeen percent at 6 sun. That gap is far too large for the waste allowance to absorb.

The footprint to enter includes the eaves overhang, not just the floor area of the building. Eaves projecting 60 cm all round add that band to the figure.

Gable and hipped roofs divide their surfaces differently, but for the same footprint and pitch they have the same total area. It is the pitch that governs, not the shape.