A roof rising 2.4 metres over a 4 metre run sits at 30.96 degrees, which is a ratio of 1 to 1.67 and a slope factor of 1.166. Over a 120 square metre footprint, the true roof surface is 139.9 square metres.
That is nearly 20 square metres of extra material for a building whose plan has not changed at all.
Three ways of saying the same thing
Pitch gets quoted as an angle in degrees, as a ratio of rise to run, or as a percentage grade. They describe one geometry, and confusion between them is a common source of ordering errors.
The ratio form is itself ambiguous, because some trades quote rise over run and others quote run over rise. The safest habit is to state both numbers with their meaning attached rather than a bare "one in three". The slope percentage calculator converts between grade, angle and ratio for the general case.
The slope factor is what you multiply by
The slope factor is the ratio between true roof area and plan area. At 30.96 degrees it is 1.166, so plan area multiplied by 1.166 gives roof area. It rises quickly as the roof steepens, and at 45 degrees it is about 1.41.
This single multiplier is the number to carry through the rest of the job, because every area based material follows it: sheeting, tiles, membrane, insulation, battens and labour priced by the square metre.
Why measuring the ground is not enough
Estimating from a floor plan gives plan area. Ordering to that figure underestimates every sloped surface on the building by the slope factor, and the shortfall grows with pitch. On the roof above it is 16.6 per cent, which on a tiled roof is a substantial pallet of tiles.
The roof pitch calculator returns the slope factor and true area alongside the angle, and the roof shingle calculator takes it through to bundles including wastage, which is a separate allowance on top, for the reason set out in the piece on ordering materials.
Rafters follow the same triangle
The rafter length is the hypotenuse of the rise and run triangle: 4.665 metres for the roof above, against a 4 metre run. Ordering timber to the run rather than the rafter length is the same mistake in one dimension instead of two.
Pitch also carries constraints beyond geometry. Many roofing products have a minimum pitch below which they are not warranted, since water moves too slowly to clear, so the practical range is often set by the material rather than by preference.