Slope, Roof, Hip Valley, Height and Width Factors

The various factors and how they are calculated.

Calculating the square footage of a roof or the length of a rafter using a factor. This factor comes with different names, roof slope factor, roof slope multiplier, roof pitch multiplier, secant, or what it most likely should be called, a slope factor.

The rise in inches per foot is a slope ratio. 6 : 12 = the slope rises 6 inches for ever 12 inches of run. You can also call this a pitch, usually written as a fraction, 6/12 reduced to, a 1/2 roof pitch. But if you want to call 6 : 12 a roof pitch, for sure you can, everyone will know what you are talking about.

Slope Factor (sf)

Used for calculating roof square footage and common rafters length.

sf * run = rafter length
(building width + eaves) * (building length + gables) * slope factor = square footage of roof

To calculate length c use Pythagoras: c = square root(run² + rise²)

c = sq rt(12² + 6²)
c = sq rt(144 + 36)
c =  sq rt(180) = 13.41640786499873817845 c = 13.42"

Now you can calculate the slope factor for a 6 : 12 roof pitch.

slope factor  = slope ÷ run
13.41640786499873817845 ÷ 12 = 1.1180339887

If you add the two calculations, square root(run² + rise²) and slope ÷ run in one formula it becomes:

No1/ sq rt(rise * rise + run * run)÷12 

which is the same as

No2 / sq rt((rise/run)² + 1)

Or if you are using pitch instead of slope

No3 / sq rt((pitch)² + 1)

Examples

No1/ sq rt(rise * rise + run * run) ÷ 12

sq rt(6 * 6 + 12 * 12 ) ÷ 12  
sq rt (36 + 144) ÷ 12 
sq rt (180) ÷ 12 
13.416407864998738178455042012388? ÷ 12  = 1.1180339887

No2/ sq rt((rise/run)² + 1)

sq rt((6/12)² + 1) 
sq rt((0.5)² + 1) 
sq rt(0.25 + 1) 
sq rt(1.25) = 1.1180339887

No3/ sq rt((pitch)² + 1)

sq rt((1/2)² + 1)
sq rt((0.5)² + 1)
sq rt(0.25 + 1)
sq rt(1.25 ) = 1.1180339887

Amazingly all the calculations end up exactly the same and the slope factor for a 6 : 12 slope is 1.1180339887

Hip/ Valley Factor (hvf)

Used for calculating the length of hip/valley rafters
hvf * run = rafter length
It is assumed the a hip or valley rafter runs at a 45 degree angle to the common rafters.
For the hip to stay at the same plane as the common rafter the amount of rise stays the same but the run length has to be increased. Using a 12 : 12 slope ratio as an example, you can calculate the distance using Pythagoras.

sq rt (12² + 12²) = 16.9705627484771405856
the run = 16.97 (rounded to 2 digits)
If we plug this into sq rt(rise * rise + run * run)÷12 formula:

sq rt(12 * 12 + 16.97 * 16.97) ÷ 12
sq rt(144+287.9809)÷12
sq rt (1431.9809)÷12
20.784150 ÷ 12 = 1.7320125 

The hip valley factor for a 12 : 12 roof slope is 1.7320125.

Calculating for 6 : 12 slope ratio:

sq rt(6 * 6 + 16.97 * 16.97) ÷ 12 
sq rt(36 + 287.9809) ÷ 12 
sq rt (323.9809) ÷ 12
17.999469 ÷ 12 = 1.7320125 =  1.500000000

The hip valley factor for a 6 : 12 roof slope is 1.5

Height Factor

Used for calculating the rafter/truss height and support posts
height factor (hf) = rise ÷ run

example calculation

hf * run = height
slope factor = 7 : 12 
rafter run = 134 inches
heel height = 6 inches
hf = 7 ÷ 12 = 0.58333333 

rafter height = 134 * 0.58333333 + 6 = 84.16 = 84 ³/₁₆

Width Factor

Used for calculating the rafter run from the height.
width factor (wf) = run ÷ rise

example calculation

wf * rise = run
slope factor = 5 : 12
height to plumb cut of rafter = 56 inches
wf = 12 ÷ 5 = 2.4
56 * 2.4 = 133.4 = 133 ³/₈"

Slope in Degrees

Useful in helping to determine angle for a bevel cut
degrees = atan ( rise/run) * 180/Π

example calculation

5 : 12 slope ratio
atan ( 5/12)*180/Π = atan ( rise/run)*180/Π = 22.62° is the plumb cut angle
and 22.62° is a 5 : 12 bevel cut.

slope ratio	slope factor	hip/valley factor	height factor	width factor	slope degrees	grade %
1 in 12	1.003466	1.416667	0.0833333333	12	4.764 °	8.3 %
2 in 12	1.013794	1.424001	0.1666666667	6	9.462 °	16.7 %
3 in 12	1.030776	1.436141	0.25	4	14.036 °	25 %
4 in 12	1.054093	1.452966	0.3333333333	3	18.435 °	33.3 %
5 in 12	1.083333	1.474317	0.4166666667	2.4	22.620 °	41.7 %
6 in 12	1.118034	1.5	0.5	2	26.565 °	50 %
7 in 12	1.157704	1.529797	0.5833333333	1.7142857143	30.256 °	58.3 %
8 in 12	1.201850	1.563472	0.6666666667	1.5	33.690 °	66.7 %
9 in 12	1.25	1.600781	0.75	1.3333333333	36.870 °	75 %
10 in 12	1.301708	1.641476	0.8333333333	1.2	39.806 °	83.3 %
11 in 12	1.356568	1.685312	0.9166666667	1.0909090909	42.510 °	91.7 %
12 in 12	1.414214	1.732051	1	1	45 °	100 %
13 in 12	1.474317	1.781463	1.0833333333	0.9230769231	47.291 °	108.3 %
14 in 12	1.536591	1.833333	1.1666666667	0.8571428571	49.399 °	116.7 %
15 in 12	1.600781	1.887459	1.25	0.8	51.340 °	125 %
16 in 12	1.666667	1.943651	1.3333333333	0.75	53.130 °	133.3 %
17 in 12	1.734054	2.001735	1.4166666667	0.7058823529	54.782 °	141.7 %
18 in 12	1.802776	2.061553	1.5	0.6666666667	56.310 °	150 %
19 in 12	1.872684	2.122957	1.5833333333	0.6315789474	57.724 °	158.3 %
20 in 12	1.943651	2.185813	1.6666666667	0.6	59.036 °	166.7 %
21 in 12	2.015564	2.25	1.75	0.5714285714	60.255 °	175 %
22 in 12	2.088327	2.315407	1.8333333333	0.5454545455	61.390 °	183.3 %
23 in 12	2.161854	2.381934	1.9166666667	0.5217391304	62.447 °	191.7 %
24 in 12	2.236068	2.449490	2	0.5	63.435 °	200 %
25 in 12	2.310904	2.517991	2.0833333333	0.48	64.359 °	208.3 %
26 in 12	2.386304	2.587362	2.1666666667	0.4615384615	65.225 °	216.7 %
27 in 12	2.462214	2.657536	2.25	0.4444444444	66.038 °	225 %
28 in 12	2.538591	2.728451	2.3333333333	0.4285714286	66.801 °	233.3 %
29 in 12	2.615392	2.800050	2.4166666667	0.4137931034	67.521 °	241.7 %
30 in 12	2.692582	2.872281	2.5	0.4	68.199 °	250 %