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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