Slope, Pitch and True Surface Area

Why the footprint is not the area

A roof measured from above covers a certain footprint. The surface itself is longer than that footprint because it rises as it runs, and the ratio between the two is fixed by the pitch alone. It is the hypotenuse of a right triangle whose base is the run and whose height is the rise.

The consequence is direct: order shingles against the footprint of a 12:12 roof and you will be forty-one percent short. The error is invisible on paper and obvious on the last day.

Pitch expressed as an angle

Pitch notation and angle are two descriptions of the same thing. A 6:12 roof is 26.6 degrees from horizontal; a 12:12 roof is exactly 45. Converting between them is a single arctangent, and it matters because clinometers and phone sensors report degrees while roofing materials are specified against rise per twelve.

What the slope factor does not capture

The factor is exact for a plane. It says nothing about hips, valleys or dormers, each of which adds surface beyond the plan projection and generates raking cuts that produce unusable offcut. That additional area is what the waste factor is doing on a roofing page, which is why a complex roof carries fifteen percent or more where a simple gable carries ten.

The factor is negligible until suddenly it is not

The slope factor is the reciprocal of the cosine of the pitch angle, and it grows slowly at first and then quickly. At a shallow four-in-twelve it is about 1.054 — five per cent, which is inside most waste allowances and is why a shallow roof forgives the error. At eight-in-twelve it is about 1.202, and at twelve-in-twelve, a forty-five degree roof, it is 1.414.

That last figure is the one worth carrying: on a square-pitch roof the surface is FORTY-ONE PER CENT larger than the plan it sits over. An order taken off the footprint is not slightly short, it is short by more than a third of itself, and no waste factor absorbs that.

The same curve explains a common site argument. Two roofers disagreeing about a quantity are often disagreeing about whether the measurement was taken on the slope or on the plan, and the gap between them is exactly this factor. Asking which way the tape ran settles it faster than re-measuring.

Hips and valleys run on a shallower slope than the roof does

A common rafter rises over a run measured straight up the slope. A hip or valley rafter rises the same height over a run measured along the DIAGONAL of the plan, which is longer by a factor of the square root of two on a regular hip.

So the hip is a shallower member than the roof it sits in: on a six-in-twelve roof, the hip is effectively six in about seventeen. Its length cannot be taken from the common rafter's slope factor, its bevel cuts are different at both ends, and a cut list that applies one factor to every rafter in the roof produces hips that are short.

Irregular hips — where the two planes meeting at the hip have different pitches — break the square-root-of-two shortcut entirely, and the run has to be constructed from the two plan dimensions rather than assumed. This is the usual reason a hip roof takes longer to set out than its area suggests.

The plan you multiply is not the building footprint

The area the slope factor converts is the plan area of the ROOF, and a roof is larger than the building under it. Eaves and verge overhangs project beyond the walls on every side, and on a small building those projections are a significant fraction of the total.

Dormers add area on their own planes and subtract almost none from the main roof, because the opening they sit in is smaller than the surfaces they add. Gable-end returns, porch roofs and lean-tos each carry their own pitch and have to be measured separately rather than folded into an average.

Openings usually are not deducted. A chimney, a rooflight or a small plant plinth removes a little material and adds cutting, flashing, detailing and waste around its perimeter, so estimating practice below a threshold size is to ignore the deduction rather than to take it. That is a deliberate convention rather than an oversight, and it is why a measured quantity can legitimately exceed the geometric one.

Where the factor is the wrong tool

The slope factor assumes a plane. A curved, conical or ogee roof has no single pitch, and its surface has to be developed from its geometry or approximated as a series of facets — which is a different calculation and, for a cone, one with a closed form worth using rather than a factor.

It also says nothing about loading, and loading moves in the opposite direction to area. As a roof steepens, the snow load acting on it falls, because a sloping surface holds less and sheds more; the wind uplift changes character; and the area of material to be carried rises. A steeper roof is more material and, often, less structural demand per unit of it.

And the factor converts area, not labour. Work on a steep roof is slower, needs more access equipment and attracts different safety provisions, so the cost factor and the quantity factor are different numbers on the same job. The pages here return the quantity, and they say that the pricing consequence of a steep pitch is larger than the geometric one.

A line across a slope sees only part of it

A wall set at an angle θ to the line of maximum fall sees the site gradient times cos θ — 87 per cent of it at 30 degrees, half at 60, none along the contours — and that is the fall a stepped foundation has to divide into steps. Each level bay between steps is the step height over the gradient along the wall.

Approved Document A laps each step by the greatest of twice its height, the foundation's thickness or 300 mm (12 in) for a strip, and by the greater of twice the step or 1 m (3 ft 3 in) for trench fill. When the bays come out shorter than the lap, every bay laps into the next and the foundation is a continuous stepped mass.