HOW TO FIND THE AREA OF A SLOPED ROOF
Using the roof pitch and the footprint of the building:
(Also see the Roof Pitch Multiplier page.)
(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, or pitch, 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 span (the horizontal span 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 over 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. So pretty standard sloped roofs have slopes like 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 uses degrees, which is easier to understand in the abstract, but it takes an extra step to figure it out mathematically, and it doesn't convey as much information on a practical level as the rise/run ratio expression). You can find the slope by getting up there and measuring with a ruler and a spirit/bubble level (or 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 your roof pitch, 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 https://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 actual dimension of the real roof surface, and you want to find out much longer that length is than the horizontal span that it covers. 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 would be 1.118 times as wide as the footprint of the roof if it were laid down flat. And for a roof with a pitch of 6/12, that number, 1.118, is your roof pitch 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.
(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 pitch multiplier for the different areas.)
As a shortcut, the roof pitch multiplier for any slope can be determined by finding the square root of ((rise/run)² + 1).