## The Slippery Slope of Pitch…

Slope is a mathematical concept that is critical to much work in Algebra and other courses. The equivalent concept in Building Construction is Pitch.

In the construction world the concept of pitch is most often applied to the angle or slope of a roof and is defined as “rise over run”. In other words the pitch of a roof is actually a ratio, expressed as a fraction, with the numerator being the rise and the denominator being the run.

So a roof that “rises” 5 inches vertically in a “run” of 12 inches horizontally, is referred to as a 5/12 pitch roof.

This seems fairly straight forward and one might assume that a storage shed roof that rises 2 feet over a run of 4 feet would be described as having a 2/4 pitch. However, many occupations employ “conventions” or standard ways of proceeding or communicating. One of the conventions regarding pitch in the construction trades is that the pitch is always converted to have a run of 12 inches. In mathematical terms this means that any pitch with a denominator other than 12 must be converted to an equivalent fraction (ratio) with a denominator of 12. Therefore in the example above, the storage shed roof with a pitch of 2/4 would be converted to a fraction or ratio of 6/12.

The Pitch of a roof may be measured by using a level and a tape measure, either from the surface of the roof or the rafters within the attic. Both methods use a level to establish the run line and the tape measure to measure the rise. Both of these procedures are illustrated in the pictures below.

Measuring pitch from roof surface: measure down to the shingles from a 12″ mark on the level.

Measuring pitch when in attic: Measure up to the rafter from a 12″ mark on the level. More information on actual measurment techniques may be found at: http://www.improvenet.com/HomeOwner/ProjectTools/Calc/roof.html

An additional convention in the construction trades is to describe the pitch by stating the numerator (rise) first and the denominator (run) second. In other words the pitch of the storage shed roof would be stated as “six-twelve” or “six in twelve” – not in standard fractional terminology, which would have been “six-twelfths”.

How then, are pitch and slope alike?

- Both terms describe the steepness of a sloped line or surface.
- Both terms are basically describing the steepness of the hypotenuse of a right triangle created with the x-axis representing run and the y-axis representing the rise.
- Construction conventions always place the rise over the run as a fraction describing pitch; mathematics conventions always place y over x as a fraction in describing slope.

How are pitch and slope different?

- Pitch is always described as a positive number; slopes can be negative or positive.
- Pitch is always converted to have a denominator or run of 12; the x factor in the slope ratio can be any real number (whole numbers, decimals, negative or positive).

Mike,

This turned into a wonderful exercise in my Residential Architecture class.

Thanks for the toughtful way you created this. It made a great lesson.

REM

RMONTE said this on November 15, 2006 at 11:41 am

Thanks so much for theinfo! I had to write a 5 page paper for my Algebra 2 class saying how slope plays a part in construction and how it affects it and how it is discovered. Thanks so much!

Betsy said this on September 25, 2008 at 9:20 pm

Mike,

I have been having the students in my Exploring Technology Class use this formula to determine the slope of ramps which they create in a Marble Maze activity. I am going to use this to explain it from now one.

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Riddle me this: if the rafter of a 12 in 12 roof runs at 45 degrees, why doesn’t the rafter of a 6 in 12 roof run at 22 and 1/2 degrees?

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