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:

*
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

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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