Triangle Slope Calculator
Enter a rise and run to find the slope of any right triangle or inclined line. Switch to the two-points mode to work directly with coordinates, or enter an angle to reverse-solve. You get the slope ratio, the angle of inclination, the slope as a percentage grade, and a full show-your-work breakdown of every step.
Formula
Worked example
A ramp rises 3 m over a horizontal run of 4 m. Slope = 3/4 = 0.75. Percentage grade = 0.75 x 100 = 75%. Angle of inclination = arctan(3/4) = arctan(0.75) = 36.87 degrees. Complementary angle beta = 90 - 36.87 = 53.13 degrees. Hypotenuse = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 m.
What is the slope of a triangle?
The slope of a triangle refers to the slope of the hypotenuse of a right triangle, which is the same as the slope of any straight line drawn between two points. A right triangle formed by a horizontal run, a vertical rise, and a slanted hypotenuse is the visual tool most often used to explain slope in geometry and surveying. The ratio rise/run gives a single number: positive for an uphill line, negative for a downhill line, and zero for a perfectly level surface. Because slope is a ratio, its value does not depend on the size of the triangle you draw, only on its shape.
How to calculate slope from rise and run
The fundamental formula is slope = rise / run, where rise is the vertical change (positive upward) and run is the horizontal change (always positive when moving in the reference direction). For example, a staircase that rises 2 metres while extending 5 metres horizontally has a slope of 2/5 = 0.4. To express the same number as a percentage grade, multiply by 100: 40%. If you know two coordinate points (x1, y1) and (x2, y2) instead, rise = y2 - y1 and run = x2 - x1, so slope = (y2 - y1) / (x2 - x1). A vertical line (run = 0) has an undefined slope, because the denominator is zero.
Angles, slope, and the right-triangle connection
The angle of inclination alpha is the angle the hypotenuse makes with the horizontal base. Because the tangent of an angle equals the opposite side divided by the adjacent side in a right triangle, tan(alpha) = rise / run = slope. You can therefore find alpha from the slope by taking the inverse tangent: alpha = arctan(slope). The complementary angle beta at the top corner of the triangle equals 90 - alpha degrees.
- Slope 0.25 = 14.04 deg (gentle road grade)
- Slope 0.577 = 30 deg (a 1-in-1.73 incline)
- Slope 1.0 = 45 deg (the steepest common ski run, also 100% grade)
- Slope 1.732 = 60 deg (very steep, uncommon in roads)
A slope of 100% does NOT mean 90 degrees. It means 45 degrees, because tan(45) = 1, and 1 x 100 = 100%. Percentage grade and angle of inclination are two different ways to express the same steepness.
Percentage grade and real-world applications
Percentage grade is widely used in civil engineering, transportation, and outdoor recreation. Road design standards in most countries limit highway grades to 6-8% for safety (grades above 6% require trucks to use lower gears). In architecture, accessible ramps under the Americans with Disabilities Act must not exceed 8.33% (1:12 ratio). Ski slopes use percentage grade to classify runs: beginner slopes are typically below 25%, intermediate slopes 25-40%, and expert slopes 40-60%. Roof pitches are often expressed as a rise-over-run ratio for a 12-unit run, so a 6/12 roof pitch equals a 50% grade and a 26.57-degree angle.
Hypotenuse and the Pythagorean theorem
The hypotenuse of the slope triangle, the actual sloped distance, is found using the Pythagorean theorem: hypotenuse = sqrt(rise^2 + run^2). This matters whenever you need to measure along the slope rather than horizontally, such as when calculating the length of a ramp, the total linear distance of a hiking trail, or the amount of fencing needed on a hillside. If slope = 0.75 and run = 4, then rise = 3 and hypotenuse = sqrt(9 + 16) = 5. Note that the hypotenuse is always longer than both the rise and the run individually.
Slope steepness categories by percentage grade
| Grade range | Category | Angle range | Typical context |
|---|---|---|---|
| 0-2% | Flat | 0-1.1 deg | Accessible ramps, parking lots, flat terrain |
| 2-8% | Gentle | 1.1-4.6 deg | Sidewalks, highway exits, mild hiking trails |
| 8-25% | Moderate | 4.6-14 deg | Steep roads, beginner ski runs, fire roads |
| 25-58% | Steep | 14-30 deg | Mountain roads, advanced ski slopes, challenging trails |
| 58-100% | Very steep | 30-45 deg | Cliff roads, extreme ski terrain, rock scrambles |
| 100%+ | Near-vertical | 45-90 deg | Cliff faces, construction grades, scaffolding |
Common classifications used in civil engineering, road design, and outdoor recreation.
Frequently asked questions
What does a slope of 1 mean?
A slope of 1 means the line rises exactly 1 unit for every 1 unit it travels horizontally, which corresponds to a 45-degree angle and a 100% grade. It is the threshold between slopes whose angle is less than 45 degrees and those steeper than 45 degrees.
How do I calculate slope from two points?
Subtract the y-values to get the rise (y2 - y1) and subtract the x-values to get the run (x2 - x1), then divide: slope = (y2 - y1) / (x2 - x1). For example, points (2, 3) and (7, 13) give rise = 13 - 3 = 10 and run = 7 - 2 = 5, so slope = 10 / 5 = 2.
Is a 5% slope steep?
No, 5% is considered a gentle slope. It corresponds to about 2.86 degrees of inclination. Most people can walk up a 5% grade without significant effort, and highways routinely use grades in this range. Slopes above about 25% start to feel noticeably steep on foot.
What is the difference between slope and angle of inclination?
Slope is the ratio rise/run, a dimensionless number. Angle of inclination is the angle the sloped surface makes with the horizontal, measured in degrees. They describe the same steepness in different units. The conversion is: angle = arctan(slope), and slope = tan(angle).
Can slope be negative?
Yes. A negative slope means the line goes downhill as you move in the positive x direction (rise is negative when run is positive). A slope of -0.5 means the surface drops 0.5 units for every 1 unit of horizontal travel. Percentage grade and angle are often quoted as absolute values, but the sign of the slope carries the direction.
What is the slope of a horizontal line?
A perfectly horizontal line has a slope of 0 because the rise is zero for any non-zero run. The angle of inclination is 0 degrees and the percentage grade is 0%. A vertical line, by contrast, has an undefined slope because the run is zero and division by zero is not defined.
How does this calculator handle downhill slopes?
Enter a negative rise in the rise-and-run mode, or use two coordinate points where y2 is less than y1. The slope will be negative, the angle of inclination will be reported as negative degrees, and the percentage grade will also be negative, all indicating a descending line. The hypotenuse is always positive because it is a length.