How Rise Over Run (Slope) Works
"Rise over run" is the everyday name for slope: how much a line moves vertically (rise) for every unit it moves horizontally (run). Given two points on a line, (x₁, y₁) and (x₂, y₂), the slope is m = rise / run = (y₂ − y₁) / (x₂ − x₁). This calculator takes those two points and returns the rise, the run, the slope, the angle of incline, and the equivalent percent grade.
Formula and method
Subtract the coordinates to get the rise (Δy = y₂ − y₁) and the run (Δx = x₂ − x₁), then divide: m = Δy / Δx. A positive slope means the line rises left to right; a negative slope means it falls; a slope of zero is a flat, horizontal line. If Δx = 0, the two points sit on a vertical line and the slope is undefined, because division by zero has no value.
Converting slope to angle and percent grade
Once you have the slope, you can express the same steepness two other common ways. The angle of incline from horizontal is θ = arctan(m), given in degrees. The percent grade — used for roads, ramps, and roof pitch — is simply the slope times 100: grade% = m × 100. For example, a slope of 0.75 is an angle of about 36.87° and a 75% grade.
Common sources of error
- Swapping the point order: always subtract in the same order for both coordinates — (y₂ − y₁) over (x₂ − x₁), not a mix of point 1 and point 2.
- Zero run: if x₁ = x₂, the line is vertical and the slope is undefined — it is not zero.
- Confusing slope with angle: a slope of 1 is a 45° angle, not a 100% or 1° incline — convert with arctan before comparing to an angle spec.
Real-world applications
- Stair and roof pitch use rise over run directly (e.g., a "6-in-12" roof rises 6 inches for every 12 inches of run).
- Wheelchair ramp design follows ADA guidance of a maximum 1:12 slope (about 8.3% grade).
- Road signs post grade as a percentage — a "7% grade" sign means a slope of 0.07.
- Graphing and algebra use slope to describe the steepness and direction of linear equations (y = mx + b).