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Pitch Illustration
Figure 1

Using Algebra and Trigonometry in Roof Framing

Of course any carpenter knows that the basis for roof framing is math. It is almost impossible to frame a roof without at least a little mathematics. Sometimes, with a little extra knowledge, the job of figuring rafter lengths, and many other things can be made faster.

Bridge Length


The single most important equation to remember in roof framing is Pythagoras theorem. With this equation, any pitch can be calculated. In roof framing the variable "B"(run) is always 12 inches. The "A"(rise) variable can be any pitch desired. The variable "C" is the "bridge length" of that particular pitch. The "bridge length" is jargon for the length of a rafter that has a run of one foot. This becomes the base number used to figure rafter lengths for different spans of that particular pitch.

An example:
To figure a simple 5/12 bridge length, you fill in the variables.
25 + 144 = 169
C = square root of 169

The bridge length for a 5/12 pitch is 13.

Now, having the bridge length, you can figure any common rafter length for a 5/12 by multiplying 13 times the run. If you had a garage which was 20 feet wide, your run is 10 feet. Multiply 10(run) by 13(bridge length of a 5/12) and you have the rafter length in inches. On a framing square which has the rafter tables, the first line, "Length of common rafter per foot of run", gives you the bridge length for the most common pitches.

Bridge Length for Hip and Valley Rafters

Figuring the bridge length for hip and valley rafters is very similar to common rafters. One detail changes, however. You have to use Pythagoras theorem twice to calculate hip and valleys. The reason is because hips and valleys depart from the wall plate on a 45 degree angle to the common rafters, which means the run is actually longer for a hip or valley than for a common rafter.

Imagine a 12 inch square piece of paper (figure 2). If you measure across it in either direction, it is 12 inches wide, but if you measure from one corner to the opposite corner the distance increases. To shorten the steps necessary to figure a hip or valley bridge length, a standard measurement of 288 inches is used, but if you use the Pythagoras equation 16.970563 inches will be the result.

To calculate the bridge length for a hip or valley you simply substitute 288 (B2)instead of the standard 144 used for common rafter calculation.

Finding angles

Now that we know the formulas for finding rafter length, we need to know what angle to cut them on. The plumb cut is the simplest to calculate, though it does require a scientific calculator. The equation is as follows:

Plumb cut miter in degrees = arctan(rise/run)

To find the angle of the slope cut, you merely deduct the plumb cut angle from 90.

Other Applications

I frequently have to use these equations for chamfered hips. These are used around bay windows and on unique home designs. Chamfered hips are simular to regular hips. The difference is that they depart from the plate at a 22 1/2 degree angle rather than the typical 45 degrees. There is usually a pair of 22 1/2 degree hips to make a chamfered hip(figure 3). To calculate a chamfered hip, you only make one variation from a normal hip. When using Pythagoras theorum to figure the diagonal run for a hip, you use :

62 + 122 = diagonal run2
The resulting 13.416408 is then used just as you would normally figure a hip.

Someone might wonder, "Why learn these equations when they have already been calculated for you on a framing square or in a rafter tables book?". My thought is this; there is very often situations in construction where the answer is not in the book. You must sometimes figure it out for yourself, and these equations put you ahead of the game. For instance, in remodeling an old house you may have to match an irregular roof pitch. While figuring a concrete work bid you may relize you need to know the distance down a sloped lot, but forgot to measure it while on site. The revered 345 method for squaring a layout is nothing but Pythagoras theorum. In the end, it is quite possible that for basic construction it is faster to use precalculated tools for your rafters, but as projects get more complicated, it is very convenient to have the necessary equations in your head.

Figure 2