When I got out of high school I had
no idea what I wanted to do for the rest of my life, besides travel and have a
good time. During the summer and school breaks I would work for a friend’s dad
who owned a successful residential construction business in the rural area
around Portland, Maine. I really didn’t have any interest in being a carpenter
and to be honest I didn’t even think that highly of most of the guys I worked
with, most of them seemed uneducated and dirty. I went to college at Montana
State University directly after high school and I signed up for general studies
but only lasted a couple semesters before I realized without a plan, what is
the point? So I dropped out and began the long road of trying to find out who I
was and what I wanted to do. I worked doing telemarketing, I was a line cook, I
did landscaping in the summer and snowplowing in the winter, and I even working
for a company cleaning and restocking toilet paper in portable toilets. Most of
these jobs paid $8-$10 per hour but the worst part was that I hated doing the
work, it really sucked. Then one day I was talking with a guy I knew, a friend
of a friend, and he asked me if I had any construction experience. After
talking for a while about it he told me to come and talk to his boss the
following Monday and I would most likely have the job (and it would start at
$14/hr.).
That Monday
I called his boss and he sounded like a great guy. He questioned me on what
experience I had and what my interest was in carpentry. To make the story short,
I got the job and worked for the company for 10 years. More importantly, work
was fun for me. I loved the owner of the company and everyone I worked with and
I loved what I was doing. Where in the world can you go be outside in the
shadows to the Rocky Mountains every day? Or to work on job sites next to Bears,
Moose, Deer, Elk…etc.?
Now is the
part that refers to this class and mathematics. My boss was an extremely smart
man, he had dual master’s degrees from Montana State in Philosophy and Mathematics,
and he would not allow calculators on his job sites. To him if you couldn’t
figure out what you needed to cut without a calculator then you couldn’t do
your job. Construction is about deadlines, and like most professions time is
money. So I will explain in depth the most common mathematics applications in
the framing of homes, how to perform the calculations, and a couple other
everyday tricks using math on a job site.
Roof
framing has an obvious use of Trig and more specifically the Pythagorean
Theorem. For simplicity I have made the house a nice even number at 20 feet
outside of wall to outside of wall. In construction instead of using angles we
use rise over run with a run of one foot. It is the carpenters job to know what
angles that is but we always carry what is called a speed square which when
used will tell you the exact angle. So the angle of the roof rafter drawn above
is called an “8,12” which has an angle of 33.75 degrees.
The
first step will be to divide the overall run of the wall the roof will sit on
by 2 (20/2 = 10ft) because you will have rafters on both sides of the building.
We now have our first length of our triangle which is 10ft. The second step
will be to calculate the height of the ridge beam that the top of our rafter
will sit against and be fastened to. The top that sits against the beam will be
where we have a 33.75 degree angle cut on our rafter. The technical name for
this cut is a “plumb cut” because in carpentry there are two types of level,
“plumb” and “level”. Plumb is level in a vertical application, such as a post
being plumb up and down. Level is on a horizontal plane like a shelf of a
counter top being level from the left to the right or vice verse.
The
second steps calculations we can take two ways. First would be more appropriate
for the purposes of this class and that is finding the length of side B. We are
able to do so because side a is tangible and we can measure it. We know it is
10ft or 120in and we know that the angle of the plumb cut or the “pitch” we
want on our roof is 33.75 degrees. We know from Trigonometry that we have the
measurement of the adjacent and the angle so we want to calculate the opposite.
We can do so because SOHCATOA tells
us to use tangent, we write down the fraction for tan 33.75 = opposite/Adjacent
= B/120 (10’ = 120”)
To solve:
Start with : B/120 = tan 33.75
Calculate tan 33.75: B/120 = 0.66818
Multiply both sides by 120: B = 0.66818 x 120
B = 80.1814
Approximately 80 3/16” in Height
Another simpler way to do this would be:
(8/12)(10)
This way would be much faster and seeing as
though ¼” is more than acceptable on something such as this the way I would go.
The
third step is to calculate the rafter length or hypotenuse but there is a bit
of a trick that makes a rafter more complex then just a hypotenuse. The rafter
has to sit on the top plate or top of wall. This is where the seat cut is and
actually combines with a plumb cut to for the birds mouth that will seat on top
of the wall, the seat cut will be the compliment of the angle in this right
triangle so if our plumb cut is 33.75 then our seat cut will be 56.25 degrees
(90 – 33.75 = 56.25). First, to calculate the hypotenuse,
A = 120
B = 80
Pythagorean Theorem = a^2 + b^2 = c^2
So, C = 144.222
C = 144 3/16”
Now that we have the triangular measurements
for the rafters, we need to know the exact locations to place those
measurements. I will explain where to
measure, what to account for, and why we account for those things.
First
we take the overall measurement that we calculated for the rafter, which is the
hypotenuse. That measurement is 144” from the top of the rafter at the plumb cut
which is 33.75 degrees. We measure the distance of 144” down to a point on the bottom of the rafter
that will sit on top of the wall plate at height measurement Y in the figure
above. From here we transfer a line down at the plumb angle of 33.75 degrees.
Now all we do is adjust for the thickness of the wall so if we have a 2x6 wall then we will adjust so that the cut
is 5.5” long at a 56.25 degrees angle. Now all that is left is to square down
at the 33.75 degrees so that the rafter runs down the outside of the wall. We
do this so that the wall carries the roof load and creates more stability. We
also do this so that the rafter “grabs the wall disabling it from moving out
or in other words enabling any lateral movement. The last this to do so that
the rafter can be test fitted is to measure half of the distance of the
thickness of your ridge beam in a parallel manner, so here we have a 1.5” ridge
beam so we measure .75” over and cut that off of the ridge cut of the rafter.
We do this because when we performed the calculations for the rafters we did so
to the center of the space. In reality the center of the space is in the middle
of a 1.5” piece of wood, the same principle would apply if the ridge beam was
3.5” wide or 5.5” wide, we would take a parallel measurement over equaling half
of 3.5” (1.75”) or half of 4” (2”).
Now
the rafter can be test fitted to insure that the math and the measurements are
correct. We do that by physically setting the rafter into place in various
locations to confirm all is uniform throughout the space.
After
we have confirmed that the rafter is the correct size and has the correct
angles, we measure the distance over for the overhang which is as it implies,
the distance the roof will overhang the wall at the outside of the building.
To do this we use a framing square which simply
is a metal stick that forms a 90 degree angle and has one inch unit
measurements in two directions in one direction it is 16” long and in the other
direction it is 24” long. If the roof is called out by the architect to
overhang one foot and we know that the roof has 8” of rise to 12” of run we
simply set one bar of the square at 12” and the other bar at 8” (on the same
edge of the rafter. We transfer over one foot and mark and cut that line which
once again is the plumb line or 33.75 degrees. We now have completed the
cutting of the rafter so we can take that rafter and transfer the lines over to
however many rafters we have figured out that we need for the space. Once they
are all cut, they are put into place, nailed off, and you have yourself a roof.
·
Being able
to use math to solve this and many other problems when building a home are
priceless. Most of the time the most valuable tool a carpenter has is his
knowledge and experience. Think about it, if you couldn’t perform the
calculations how would you build a roof on a home? Would you just guess?
Probably not because if you did that you would have to use trial & error
which would take you days if not longer to do what you could do in a few hours
using math. Carpenters use math all the time, other priceless pieces of
knowledge for a carpenter are using 3-4-5 to find square, and calculating area
and volume when ordering materials or estimating. I have used formulas on job
sites to calculate the circumference of a circle (C = 2πr), a radius with a
chord dimension ((Radius = 1/2 (rise2 + 1/4 chord2) / rise),
and many others.
