Roof Slope Factor:
How to Find The Area of a Sloped Roof
Using the Roof Slope and the Footprint of the Building
Roof Online Staff
Scroll to the bottom if you just want to see the handy chart.
Determine the area of the roof’s footprint, which is the area covered by the roof regardless of slope (or the apparent area of the roof when you look straight down at it, like from a satellite). Google Earth makes this pretty easy, but you can also simply measure the dimensions of the building on the ground and figure out the area that way. (See here for help using Google Earth.) Don't forget to take into account overhanging eaves or any other areas where the roof extends beyond the exterior walls of the building. You will also need to increase your total footprint area to include any areas where one roof section overhangs another roof section.
Determine the slope (commonly called the “pitch”, although slope and pitch are not actually the same thing) of the roof. Please note that all of these calculations work the same whether you are using inches or centimeters. You just have to plug the numbers in. A roof that rises 10 cm for every 30 cm of horizontal distance (the horizontal distance is called the run) has the same slope as a roof that rises 4 inches for every 12 inches of run.
In the US, the slope of the roof is typically expressed as a ratio, "rise-in-run", or "rise:run", which by general agreement in the US roofing industry always uses 12 inches as the run, and states how many inches the roof rises vertically for every 12 inches the roof goes across horizontally. Sloped roofs commonly have slopes like 6-in-12 (6:12) or 8:12. A roof that goes up at a 45-degree angle has a slope of 12:12.
(The rest of the English-speaking world simply uses degrees, which is easier to understand in the abstract, but it takes a few extra steps to figure it out mathematically, and it doesn't convey as much information about the roof on a practical level as does the rise:run ratio expression. To find the slope in degrees, you divide the rise by the run and find the inverse tangent of the result. But you still need to have the rise and the run to find the degree angle if you don’t already know it. If you do know the roof slope in degrees, then all you have to do is find the secant using a scientific calculator. For example, if the roof slope angle is 45°, then sec(45) = 1.414213. There's your roof slope multiplier.)
You can find the slope by getting up there and measuring with a ruler and a spirit level, or you can use a specifically-designed slope finder.* You could also take your measurement in the attic using a rafter, the underside of the roof deck, or even the ceiling, if you’re sure that it follows the slope of the roof. Eyeballing the slope and guessing (like "well, it looks like it goes up 6 feet for every 12 feet it goes across, so it's about a 6:12 slope) is probably not a good idea. The result you get from guessing like this might be close enough for government work, but if it’s a large roof and you’re a roofer with a 10% profit margin, I wouldn’t recommend it.
There are also a number of smartphone apps that you can use to find the slope of your roof, if you have a clear view of a gable end on the building. These can be acceptably accurate sometimes. You don't have to get up on the roof or go to the attic to use these. Go search your app store for "roof pitch" apps. You'll see.
Once you have the slope of the roof, you can consider yourself in possession of the lengths of the a and b sides of a right triangle. Now you need to pull out your trusty Pythagorean Theorem and figure out the length of the hypotenuse. A convenient calculator can be found at web2.0calc.com.
As I’m sure you remember, a² + b² = c². So, if your slope is 6:12,
6² + 12² = c²
36 + 144 = c²
180 = c²
√180 = c
13.416 = c
So 13.416 is the length of the hypotenuse.
Now that you have the length of your hypotenuse, 13.416, you want to think of that as the length of the amount of the actual roof surface that covers a horizontal distance of 12 inches. You want to find out how much longer that length is than the 12 inches, and express the difference as a number, the roof slope factor (also called the roof slope multiplier, roof pitch factor, or roof pitch multiplier). Once you know the roof slope multiplier for a particular slope, you can use it to find the actual area of any size sloped roof with the same slope.
So in this case, for every 12 inches the roof goes across horizontally (b), while rising 6 inches (a), there’s actually 13.416 inches of roof (c). If you divide the 13.416 inches by the 12 inches (c/b), that gives you 1.118. So the actual roof surface, if it were laid down flat on the ground, would be 1.118 times larger than the footprint of the roof. And for a roof with a slope of 6:12, that number, 1.118, is your roof slope multiplier.
Take the area of the roof’s footprint, say it's 5,000 square feet, and multiply 5,000 by 1.118. That gives you 5,590 square feet, which is the actual area of the surface of the roof. And that’s the number you were looking for.
This applies to lengths as well as areas. A rafter for a roof with a 6:12 slope will be 1.118 times as long as the length that it spans horizontally.
(Note that if the roof has areas with different slopes, as on a gambrel roof, for instance, you'll have to figure out more than one slope and use the appropriate roof slope multiplier for the different areas.)
As a shortcut, the roof slope multiplier for any slope can be determined by finding the square root of ((rise/run)² + 1). (Divide the rise by the run. Square the result. Add 1. Find the square root of that result.)
If you’re using the table below, you should consider getting yourself a construction calculator. That way you can instantly calculate the factor for any slope, among many other useful things. This one is very good.*