Spike's Calculators

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 %