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### Roof Rafters

```
Date: 02/03/97 at 18:51:26
Subject: Roof Rafters

Hello,

I am a general contractor. I am working with some estimating software
and setting up a database to find lumber quantities for projects I
bid.  My problem is that I am trying to figure out a formula for some
roof framing.

PROBLEM:

I am trying find a formula that will give me the length of a roof
rafter by using the pitch of the roof. All sloped roofs have what's
called a pitch. The pitch is how far the roof rises for every 12" of
horizontal run. Example: a 4/12 pitch means that for every 12" of
horizontal run, a roof will rise 4". A 6/12 pitch means it will rise
6". In a right triangle, the length of the hypotenuse would be the
same thing as the length of the rafter. The pitch would give you the
angle at the bottom of the rafter. A 4/12 pitch would mean that the
angle is 15 degrees. A 6/12 would have a 22.5 degree angle.

I know I can figure this using the Pythagorean theorem but I am trying
to devise a formula so one can use the dimensions already noted on the
blueprints without having to figure out the rise. All blueprints give
the pitch of a roof and the run will always be 1/2 the length of the
wall underneath.

I'm fairly sure that this formula will require the use of cosines
and/or sines. The unfortunate part is that my estimating software
does not have these functions so I am trying to come up with a formula
that does not require this.

Your help in this matter would be greatly appreciated.  If you cannot
help, maybe you could at least lead me in a direction that can.

Thank you very much,
```

```
Date: 02/04/97 at 10:17:14
From: Doctor Robert
Subject: Re: Not a student but needs help

Let L be the length of the underneath wall, and let R be the length of
the rafter you need. Let p be the pitch, which is the rise over the
run of the rafter. If you imagine a right triangle with the vertical
leg of length a, the horizontal leg of length b, and R the hypotenuse.
Then:

R^2 = a^2 + b^2

= b^2(a^2/b^2 + 1)

= (L/2)^2(p^2 + 1)

R = L/2 * sqrt(p^2 +1)

This should give you the length of the rafter.

-Doctor Robert,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 07/25/97 at 10:29:25
From: Doctor Ken
Subject Re: Not a student but needs help

Here's a note from the archivists.  Wayne McLennan, a reader of our
archives, just wrote in to say that his definition of pitch differs
from the original poster's.  According to Wayne, pitch is the ratio of
the total rise of the roof to the total span of the building.  Thus
the pitch is half of what mathematicians call the slope.

If this is the case, then the following modifications to Rob's work
are in order:

R^2 = a^2 + b^2

= b^2(a^2/b^2 + 1)

= (L/2)^2((2p)^2 + 1)

R = L/2 * sqrt(4p^2 + 1)

In my search of the web for information on roof pitch, I found that
there are several different standards.  Some people seem to
communicate pitch by simply noting the angle - for instance, a pitch
of 60 degrees.  Other people seem to say things like "a 10 inch
pitch", meaning that for every 12 inches of horizontal distance, the
roof rises 10 inches.  And some people express pitch as a ratio (the
same way as the original poster), saying things like "a pitch of 2 in
12".  See http://newberry.deschutes.org/CDD/planning/old/Maho.htm#5   .

-Doctor Ken,  The Math Forum
```
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

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