TI-84 Plus CE: Rafter Multipliers and Roof Area
Introduction
Rafter multipliers are a quick way to calculate the area of a roof. All that is needed are three measurements:
(1) The width of a house
(2) The length of a house, including eaves and lengths of overhangs
(3) The pitch of the roof. The pitch the rise of the roof in inches over a run of 12 inches.
It is assumed that we have a simple roof and the area of the house is rectangular. Even in more complex cases, we can use the multipliers come up with a good approximation.
The area is approximated by:
roof area ≈ length * width * rafter factor
The common values are shown here:
Pitch (n in 12”) |
Pitch (decimal – 3 places) |
Rafter Factor (decimal -3 places) |
2 in 12 |
0.167 |
1.014 |
3 in 12 |
0.250 |
1.031 |
4 in 12 |
0.333 |
1.054 |
5 in 12 |
0.417 |
1.083 |
6 in 12 |
0.500 |
1.118 |
7 in 12 |
0.583 |
1.158 |
8 in 12 |
0.667 |
1.202 |
9 in 12 |
0.750 |
1.250* |
10 in 12 |
0.833 |
1.302 |
11 in 12 |
0.917 |
1.357 |
12 in 12 |
1.000 |
1.413 |
13 in 12 |
1.083 |
1.474 |
14 in 12 |
1.167 |
1.537 |
15 in 12 |
1.250 |
1.601 |
16 in 12 |
1.333 |
1.667 |
17 in 12 |
1.417 |
1.734 |
18 in 12 |
1.500 |
1.803 |
19 in 12 |
1.583 |
1.875 |
20 in 12 |
1.667 |
1.948 |
21 in 12 |
1.750 |
2.010 |
22 in 12 |
1.833 |
2.083 |
23 in 12 |
1.917 |
2.167 |
* This was erroneously labeled as 1.357 on the Dewalt Construction Math Quick Check guide (see Sources). Several sources correctly has the value of 1.250.
Curve Fitting
What if I can fit a formula to the above data?
I used a TI-84 Plus CE to assist me with this.
First, we’ll use the quadratic regression with the data from the above table:
y = a * x^2 + b * x + c
The resulting coefficients:
a = 0.1867080745
b = 0.2897396951
c = 0.938
r^2 = 0.9992015789
That’s a pretty good fit.
What about the square root fit:
y = √(a * x^2 + b)
If we square both sides, we get:
y^2 = a * x^2 + b
This equation takes somewhat the form of the linear regression.
Let z = x^2 and t = y^2, then:
z = a * t + b
Pitch (n in 12”) |
X = Pitch (decimal – 3 places) |
Z = X^2 (approximate) |
Rafter Factor (decimal -3 places) |
T = Y^2 (approximate) |
2 in 12 |
0.167 |
0.0278 |
1.014 |
1.0282 |
3 in 12 |
0.250 |
0.0625 |
1.031 |
1.063 |
4 in 12 |
0.333 |
0.1111 |
1.054 |
1.1109 |
5 in 12 |
0.417 |
0.1736 |
1.083 |
1.1729 |
6 in 12 |
0.500 |
0.25 |
1.118 |
1.2499 |
7 in 12 |
0.583 |
0.3403 |
1.158 |
1.341 |
8 in 12 |
0.667 |
0.4444 |
1.202 |
1.4448 |
9 in 12 |
0.750 |
0.5625 |
1.250* |
1.5625 |
10 in 12 |
0.833 |
0.6944 |
1.302 |
1.6952 |
11 in 12 |
0.917 |
0.8403 |
1.357 |
1.8414 |
12 in 12 |
1.000 |
1 |
1.413 |
1.9966 |
13 in 12 |
1.083 |
1.1736 |
1.474 |
2.1727 |
14 in 12 |
1.167 |
1.3611 |
1.537 |
2.3624 |
15 in 12 |
1.250 |
1.5625 |
1.601 |
2.5632 |
16 in 12 |
1.333 |
1.7778 |
1.667 |
2.7789 |
17 in 12 |
1.417 |
2.0069 |
1.734 |
3.0068 |
18 in 12 |
1.500 |
2.25 |
1.803 |
3.2508 |
19 in 12 |
1.583 |
2.5069 |
1.875 |
3.5156 |
20 in 12 |
1.667 |
2.7778 |
1.948 |
3.7947 |
21 in 12 |
1.750 |
3.0625 |
2.010 |
4.0401 |
22 in 12 |
1.833 |
3.3611 |
2.083 |
4.3389 |
23 in 12 |
1.917 |
3.6736 |
2.167 |
4.6959 |
Running the linear regression calculation with (z, t) we get:
a = 1.000120031
b = 1.00008392
r^2 = 0.9999332927
More accurate that the quadratic regression.
z = 1.000120031 * t + 1.00008392
y^2 = 1.000120031 * x^2 + 1.00008392
y = √(1.000120031 * x^2 + 1.00008392)
Analysis
I suspect that the exact formula for the rafter factor is:
rafter factor = √( pitch^2 + 1 ), pitch as a decimal
And the roof area is:
roof area = length * width * √( pitch^2 + 1 ) [ in inches ]
Roof area in square feet = Roof area in square inches / 144
Example
Area: length = 18 feet 6 inches, width = 20 feet, pitch = 7/12
In inches, length = 222 inches, width = 240 inches
roof area = 222 * 240 * √((7/12)^2 + 1) ≈ 61682.45131 in^2 ≈ 428.3503 ft^2
Sources
Prince, Christopher DeWALT Construction Math Quick Check: Extreme Duty Edition. Delmar, Cengage Learning: Cliffon Park, NY. 2010
Elliot. “Roof Area Calculator * Surface Area Multiplied by Pitch”. sktechandcalc.com. October 20, 2017. https://www.calculator.net/roofing-calculator.html?acarea=176&acareaunit=feet&roofpitch=4&angle=25&eaves=0&eavesunit=feet&price=&priceunit=feet&ctype=pitch&tp=ar&x=Calculate
calculator.net. “Roofing Calculator” 2008 – 2024. Retrieved November 4, 2024. https://www.calculator.net/roofing-calculator.html?acarea=176&acareaunit=feet&roofpitch=4&angle=25&eaves=0&eavesunit=feet&price=&priceunit=feet&ctype=pitch&tp=ar&x=Calculate
Eddie
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