Roof take-off: slope areas, ridges, verges and the pitch factor

This guide follows French practice: French standards (NF DTU), French regulations and French bill conventions. Most sources are in French.

Roof items and their unit

A few words are enough to describe a roof. A slope (or roof plane) is a flat, inclined surface. The ridge is the high, horizontal line where two slopes meet; the eaves are the low line, also horizontal, which carries the gutter. The verge is the side edge of a slope, rising along a gable. A hip is the external edge between two slopes (on a hipped end), a valley the internal angle where water collects. Finally, the rafter length is the true length of a slope, from eaves to ridge.

Why the plan does not give the true area

A roof plan is a view from above: each slope is drawn in horizontal projection. An inclined slope is larger than its shadow on the ground; measuring the plan without correction therefore underestimates the roofing, and all the more so as the pitch gets steeper. Horizontal lines, the ridge and the eaves, keep their true length on the plan.

The pitch factor

A roof pitch is expressed as a percentage (the rise per 100 of horizontal run) or in degrees. With the pitch p written as a ratio (40% = 0.40) and the angle α, the factor that converts a plan length or area to its true size is:

factor = √(1 + p²) = 1 / cos α

This is Pythagoras applied to the rafter: for 1 m of horizontal run, you rise p metres, and the sloping length is √(1 + p²). For a 35% pitch you get 1.059; for 100% (45°), 1.414.

* Two slopes of the same pitch meeting at a right angle on plan.

Two precautions. Do not confuse percentage and degrees: a 45% pitch (24.2°) has nothing to do with 45° (100%). And keep three decimal places: rounding 1.077 to 1.08 adds 0.3%; on 2,000 m² of roof on plan, that is almost 6 m² of roofing that does not exist.

Hips and valleys

A hip does not follow the line of steepest slope. On plan it runs at an angle, at 45° when two slopes of the same pitch meet at a right angle: for the same rise it covers more horizontal distance than a rafter, and its own pitch is shallower. Its factor is √(1 + p²/2), i.e. 1.039 for a 40% pitch, against 1.077 for the slope. Applying the slope factor to hips overstates them. The valley, the mirror image of the hip, follows the same rule.

For two slopes of different pitches, the formula no longer holds. The safest way is to go back to Pythagoras: true length = √(plan length² + rise²), the rise being the difference in height between the two ends of the hip or valley.

What you count, and how

The slopes

Measure each slope on the roof plan, overhangs included if the plan shows them: check that the eaves line drawn is that of the roofing and not the face of the wall. Then multiply by the factor for that slope's pitch. Laps between tiles or slates are not added to the area: they are taken into account in the material consumption per m² (number of units according to the gauge). Openings, roof windows or chimney stacks, are deducted or not according to the contract's rule.

Ridges, eaves and gutters

Being horizontal, they are measured directly on the plan. The ridge of a hipped roof is shorter than the building; the eaves run all round the slopes. Gutters follow the eaves; downpipes are counted per item, their length taken from the elevations or sections.

Verges

A gable verge rises with the slope: its plan length is multiplied by the slope factor, exactly like the rafter.

Flat roofs

A flat roof is measured on plan: its fall is slight and its factor practically equal to 1. Upstands are counted per linear metre along parapets, roof plant rooms and penetrations, with the upstand height set out in the specification; the perimeter of each area gives most of it.

Worked example

A house of 12.00 × 8.00 m overall, overhangs included, with a dual-pitched roof at 40%, the ridge parallel to the long side. Factor: √(1 + 0.40²) = √1.16 = 1.077.

With a factor rounded to 1.08, you would get 103.68 m²; without a factor, 96.00 m², i.e. more than 7 m² of roofing forgotten.

The same house, this time with a hipped roof of four slopes at 40% (two hipped ends). The roofing area does not change: all slopes have the same pitch, so 96.00 × 1.077 = 103.39 m². It is the lines that change. The rise is 4.00 × 0.40 = 1.60 m.

The hip factor gives the same result: 5.657 × 1.039 = 5.88 m. With the slope factor, you would have written 4 × 5.657 × 1.077 = 24.37 ml, i.e. 0.85 ml too much.

The most common mistakes

• The roof plan area carried straight into the bill of quantities.
• The factor applied to ridges and eaves, which are horizontal.
• The slope factor applied to hips and valleys.
• A percentage taken for degrees, or the reverse.
• Overhangs forgotten, or counted twice when the roof plan already shows them.
• A single pitch for a roof that has several (sprocket, or the lower and upper slopes of a mansard roof).
• One item for several materials.

For workmanship (minimum pitch, laps, fixings), the French reference is the NF DTU of the 40 series for the material: for example 40.11 for slate, 40.21 for interlocking or sliding clay tiles, 40.35 for profiled steel sheets. The minimum pitch allowed depends on the material and on site conditions.

Doing it in Kotelys

Kotelys measures the drawing as it is drawn, in horizontal projection, then applies the pitch you give the type: the quantity becomes the true size, and the take-off calculations write it out (“42.10 m² × 1.118 (pitch 50%) = 47.07 m²”).

• Each slope is drawn with the Area tool, one type per material and per pitch (tiles, slates, zinc, metal sheeting, waterproofing), each with its colour and total. The pitch is entered as a percentage or in degrees, in the type's Properties (“Quantity calculation” section).
• Another method, for a rectangular slope: measure the true rafter length on a scaled section, then draw the eaves as a Length with a height equal to that rafter length. Kotelys then writes “12.00 ml × 4.31 m = 51.72 m²”.
• Ridges, eaves and gutters are drawn as Lengths, with no pitch: they are correct as they are. Verges take the pitch of the slopes; hips and valleys have their own type, with the pitch of the hip itself, entered by hand: for two slopes of the same pitch meeting at a right angle, the slope pitch divided by √2 (40% → 28.3%).
• For a flat roof, the perimeter of each area is written in the take-off calculations and in Excel: it is the basis for the length of upstands. Roof plant rooms and voids are drawn as deductions.
• Roof windows, vents and downpipes are counted with the Count tool.
• The roof plan, the sections and the elevations live in the same project, each at its own scale. The PDF take-off calculations write out every calculation; the Excel workbook gives the summary and the object-by-object detail.

Further reading: roofing and waterproofing take-off in Kotelys, and the other guides.

Sources

1. Éduscol STI, BTS EEC U42 marking scheme: roofing area and pitch factor (35% = 1.05948)
2. Habitatpresto, « Calculer la surface d'une toiture »
3. Batirama, « La liste des DTU à jour » (40 series, roofing)
4. AFNOR, NF DTU 40.13 « Couverture en ardoises en fibres-ciment » (example from the 40 series)

Thresholds and values described as “taken for the example” are not rules: those of your contract (specification, bill of quantities) always prevail.