How do I work out roof pitch and roof area?
Pitch is the angle of the slope: arctan(rise / run). True roof area is the plan area divided by the cosine of the pitch, because a sloping surface is bigger than the flat footprint under it.
Pitch from rise and run
Run is the horizontal distance from the eaves line to the ridge. On a symmetrical duo-pitch roof that is half the span. Rise is the vertical height from the eaves line up to the ridge. Then:
- pitch angle = arctan(rise / run)
- rise = run x tan(pitch)
Worked example. A 9 m span gives a run of 4.5 m, and the rise is 2.0 m. Rise over run is 0.444, so the pitch is arctan(0.444) = 23.96 degrees, call it 24.
As a ratio, the same roof rises 2.0 m for every 4.5 m of run, which is 1 : 2.25. A 1 in 3 slope works out at 18.4 degrees. Going the other way, a 35 degree pitch over the same 4.5 m run needs a rise of 4.5 x tan 35 = 3.15 m.
Slope (rafter) length
The length along the slope, from ridge to eaves line, is the hypotenuse of the triangle formed by run and rise. Either method gives the same answer:
- Pythagoras: length = square root of (run squared + rise squared)
- Trigonometry: length = run / cos(pitch)
Using the 4.5 m run and 2.0 m rise: the square root of (20.25 + 4.00) is 4.92 m. And 4.5 / cos 23.96 degrees is also 4.92 m. This is the theoretical length to the eaves line. A rafter you are cutting also needs the overhang and the joint details from the drawing.
Roof area of a duo-pitch roof
Roof area = plan area / cos(pitch). The plan area is the footprint measured flat, as on a floor plan. Dividing by the cosine gives the sloping surface. Multipliers (1 / cos) for common pitches:
- 30 degrees: 1.155
- 35 degrees: 1.221
- 40 degrees: 1.305
- 45 degrees: 1.414
Worked example. A building 9 m x 12 m has a plan area of 108 m2. At 35 degrees the roof area is 108 / cos 35 = 131.8 m2. Check by the other route: the slope length is 4.5 / cos 35 = 5.49 m, and two slopes of 5.49 m x 12 m make 131.8 m2.
Overhangs. Measure them horizontally and add them to the plan before dividing. With a 0.3 m eaves overhang each side and a 0.15 m verge at each end, the plan becomes 9.6 m x 12.3 m = 118.08 m2, and the roof area is 118.08 / cos 35 = 144.1 m2. If the drawing gives an overhang measured along the slope, convert it to a horizontal distance first (multiply by cos of the pitch).
Hips and valleys
If every plane of the roof has the same pitch, the same rule covers the whole roof, hips and valleys included: take the plan area of the whole outline, with overhangs, and divide by cos(pitch). If the planes differ in pitch, or the plan corners are not right angles, work out each plane on its own.
Hips and valleys need their own lengths, for hip tiles, valley trough or lead, and rafters. Where two equal-pitch planes meet at a right angle on plan, the hip or valley is longer in plan than the common rafter by a factor of the square root of 2. With a 4.5 m run and 2.0 m rise, the plan length is 4.5 x 1.414 = 6.36 m, and the true length is the square root of (6.36 squared + 2.0 squared) = 6.67 m.
Take dormers, chimneys and large openings as separate items.
Minimum pitch and ordering
The lowest pitch a covering can be laid at comes from the manufacturer of that tile, slate or sheet, and it can depend on the rafter length, the exposure and the lap. Check their technical data and the design before ordering.
The manufacturer also gives the coverage per square metre. Cuts at hips, valleys and verges create waste, and how much to allow is a judgement for the roof in front of you. Ridge and hip pieces are ordered by length.