In the United States roof pitch is usually represented by the numbers (in inches) we use on a framing square to mark the pitch. For instance if we were framing a house with a 7/12 pitch roof we’ll use the numbers 7 on the tongue of the square and 12 on the blade to mark the angle. We’ll also want to know what angle to set the saw so we can cut the gable studs; which will also be the angle of incline of the roof. In this case the 7 and the 12 represent two sides of a triangle with the hypotenuse being the third side. We’ll use a calculator with trigonometric functions for this.
Tangent is the trigonometric function we use to figure this out. Tangent is the number we get when we divide the opposite side by the adjacent side. Since we are carpenters we’ll call this rise divided by the run or rise/run. For example on a 7/12 roof we divide 7 by 12 to get the number .583333. This number is called the tangent. Next we’ll enter inv tan .5833 and exe on our calculator. The result is 30.255012 or 30 ¼ ˚. This is the angle of the roof (incline from horizontal). Bear in mind that this number is the decimal equivalent which might be good enough for framing but we’ll want to know how to change it to degrees, minutes and seconds.
Circle = 360 degrees, written as 360˚
Degree = 60 minutes, written as 60’
Minute = 60 seconds, written as 60”
So to convert 30.255012 degrees to (degrees, minutes, seconds) we subtract the integer portion and multiple the decimal portion by 60. For ex. (30.255012 – 30) x 60 = 15.3007 minutes. Now subtract 15 and multiply .3007 x 60 to get 18 seconds. Final answer 7/12 = 30˚15’18”.
This conversion process is an essential mathematical skill so it’s important to learn. Having said that a lot of calculators have a button on it to do this conversion automatically.
Study the diagram below.
Hypotenuse = the long side of a right triangle
Adjacent Side = Run
Opposite Side = Rise
/ = divided by
The most common angles for roof pitches are printed on the side of a speed square. However; knowing this basic mathematical formula is essential when working with complex roof designs. Notice that angle A is the slope of the roof from horizontal, the angle of the plumb cuts and the angle to set your saw when cutting gable studs for instance.
Just a quick side note here* – rise divided by the run is also how we figure percent of slope. So the formula for percent of slope is rise/run x 100 = percent of slope. In the above example percent of slope is 58. Percent of slope is often used to express road grade conditions. You’ll not likely find the term used for home building.
Ok, back to framing. Check out the handy conversion chart below.
Hip rafters are different. When determining the angle of a hip use 16.97 instead of 12 in the formula above.
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