All roofers use trigonometry, but roofing terminology sometimes obscures this fact. Instead of expressing roof pitch, which is a measure of the slope of a roof, as an angle, roofers prefer to express it as a ratio. The ratio still describes an angle, and you can convert between the ratio and the angle of the roof – as measured from the eave – using a mathematical table called a tangent table. There's usually no need to do this, however, because the ratio is self-explanatory and, in many ways, easier to use.
The Rise Over the Run
You'll often hear roofers talking about the rise and run of a roof. The rise is the height of the roof above the top of the walls and the run is the horizontal distance from the peak to the edge of the eave. For example, a roof with a peak that rises 8 feet above the top of the walls has a rise of 8 feet. If the horizontal distance from the peak to the eave is 24 feet, the roof has a run of 24 feet. The ratio of the rise to the run is then 8/24, or 1/3.
Roofers like to express the roof pitch ratio in inches, and for simplicity, they always use a run of 12 inches. To express a slope of 1/3 as a roof pitch, you would multiply by 4/4 to produce 12 in the denominator of the ratio. The fraction 4/4 equals 1, so it doesn't change the value of the pitch, and it produces a pitch of 4/12.
To reiterate, a roof pitch of 4/12 means that the roof rises 4 inches for every 12 inches of horizontal distance. Roofers refer to this as a 4-in-12 roof.
Measuring Roof Pitch
In practice, it's unusual to measure the entire rise and run of a roof. It's much easier to measure the change in roof elevation over a 12-inch horizontal distance. You can do this with a 12-inch level and a tape measure, either working on top of the roof or in the attic. You extend the level out from the roof deck toward the downward slope, center the bubble and measure the vertical distance between the decking and the free end of the level. The pitch of the roof is simply the ratio of this number to 12.
Roof Pitch and Roof Angle
You can think of a roof as forming the hypotenuse of a right-angled triangle. The rise and run of the roof form the other two sides of the triangle. Dredging up knowledge you gleaned in high school math class, you'll remember that the ratio of the rise to the run correspond to the tangent of the angle formed by the roof deck and the roof trusses. The larger this angle is, the greater the pitch of the roof. If you want to know the value of the angle, simply convert the roof pitch to a decimal fraction by dividing the run into the rise and look up that fraction in a table of tangents.