Triangle Slope / Grade / Pitch Calculator
Roofs, ramps, and roads all measure the same right triangle — they just talk about it differently. Pick your context for the numbers that actually matter to you.
Enter rise per 12 inches of run — the standard X:12 notation.
A roofer says “6:12.” A civil engineer says “1 in 20.” An accessibility inspector says “8.33%.” Three completely different sentences — and every one of them is describing the exact same thing: a right triangle, tilted up from flat ground, measured by how much it rises for how far it runs.
Once you see that, the whole topic gets simple. Rise and run are the two legs of a right triangle. Everything else — pitch, grade, ratio, angle — is just a different way of describing the same triangle to a different audience.
Rise, Run, and the One Idea Behind Every Slope
Rise is how far up something goes. Run is how far across it goes to get there. Divide rise by run, and you get the slope — a single number that describes the steepness, no matter what field you’re working in.
\(\text{slope}=\frac{\text{rise}}{\text{run}}\)
From that one ratio, every format below falls out directly.
Roof Pitch: The X:12 Standard
Roofers measure pitch as inches of rise for every 12 inches of run — written as a ratio like 6:12 or 4:12. A 6:12 roof climbs 6 inches for every foot it runs horizontally.
From that ratio, three more numbers follow: the angle, the percent grade, and something roofers call the slope multiplier — how much bigger the sloped roof surface is than the flat building footprint underneath it.
\(\text{Multiplier}=\sqrt{1+\left(\frac{\text{rise}}{12}\right)^2}\)
Worked Example
A 6:12 roof: percent grade is \((6\div12)\times100=50\%\). Angle is \(\arctan(6/12)=26.57°\). Multiplier is \(\sqrt{1+0.5^2}=1.118\).
If the house has a 1,500 sq ft footprint, the actual roof surface is \(1500\times1.118=1,677\) sq ft — 177 square feet more material than the footprint alone would suggest. That gap only grows on steeper roofs.
Ramp Grade: Building to ADA Compliance
Ramps aren’t measured for materials — they’re measured for safety, and there’s a hard legal number attached: the Americans with Disabilities Act caps a commercial or public ramp’s running slope at 1:12, or 8.33%. That works out to roughly a 4.8° incline — gentle enough for a wheelchair user to manage without excessive effort.
The practical question is almost always the same: given a rise I need to overcome, how much horizontal run do I need? At the ADA maximum, the answer is simple — multiply the rise by 12.
Worked Example
A doorway sits 21 inches above ground level. At the ADA maximum 1:12 ratio, the ramp needs at least \(21\times12=252\) inches of run — 21 feet.
That 21 feet is the horizontal distance, not quite the same as the ramp’s actual surface length. The true sloped distance, via the Pythagorean theorem, comes out to 252.9 inches — about an inch longer. At this shallow an angle the difference barely matters, but it’s the more precise number, and steeper ramps make that gap bigger.
Residential ramps have more flexibility — up to 2:12 is a common home standard, and 3:12 shows up in tight spaces, though steeper ramps get noticeably harder for powered wheelchairs and scooters to manage.
Road and General Slope: Percent Grade and “1 in N”
Civil engineers and road signage mostly use percent grade — an 8% grade road climbs 8 feet for every 100 feet of horizontal distance. You’ll also see the same information written as “1 in N” — a 5% grade and “1 in 20” are the identical slope, just written two different ways.
\(\text{1 in }N=\frac{\text{run}}{\text{rise}}\)
This mode is the most general of the three — no assumed 12-inch run, no regulatory ceiling, just the raw relationship between whatever rise and run you actually have. It works equally well for a driveway, a hiking trail, or a drainage pipe’s fall.
Converting Between Formats — Without Losing the Plot
Since every format describes the same right triangle, converting between them is just re-expressing the same ratio:
- Ratio (X:12 or 1:N) to percent: divide, then multiply by 100.
- Percent to angle: divide by 100, then take the arctangent.
- Any format to any other: find the rise/run ratio first, then convert from there.
The calculator above does this instantly for whichever context matches your project, but the underlying idea is the same triangle every time.
Frequently Asked Questions
1:12, or 8.33% grade — about a 4.8 degree incline. For every inch of vertical rise, the ramp needs at least 12 inches of horizontal run. Residential ramps sometimes use steeper ratios like 2:12, but that isn’t compliant for commercial or public spaces.
Divide the rise by the run (12, for standard X:12 notation), then take the arctangent of that number. A 6:12 pitch works out to about 26.57 degrees.
Multiply the flat footprint area by the roof’s slope multiplier: the square root of 1 plus the pitch ratio squared. A 6:12 roof has a multiplier of about 1.118, meaning the sloped surface is roughly 12% larger than the footprint.
It means the surface rises 1 unit for every 20 units of horizontal distance — the same as a 5% grade. ‘1 in N’ and percent grade describe the identical slope, just written differently.
At the ADA maximum of 1:12, multiply the rise by 12 to get the minimum horizontal run in the same units. A 21-inch rise needs at least 252 inches (21 feet) of run.
Bottom Line
Roof pitch, ramp grade, and road slope all sound like separate topics, and every calculator that treats them separately makes them feel that way. They’re not — rise over run is rise over run, whether it’s holding up shingles or a wheelchair. Pick the mode that matches your project, and the same simple triangle handles the rest.