Calculators5 min readUpdated: 2026-09-14
What Is Slope in Mathematics? Rise Over Run Definition, Incline & Pitch
Definition
Slope is a mathematical measure of the steepness and direction of a line, defined as the ratio of vertical change (rise) to horizontal change (run).
Detailed Technical Breakdown
In Cartesian geometry, the slope m of a non-vertical line passing through points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1). A positive slope rises from left to right, a negative slope falls, a horizontal line has a slope of zero, and a vertical line has an undefined slope. In real-world engineering, slope is expressed as an angle of inclination (arctan m), grade percentage (m × 100%), or construction roof pitch (rise per 12 inches of run).
Key Technical Specifications
- Slope formula: m = Δy / Δx = (y2 - y1) / (x2 - x1).
- Angle of incline: θ = arctan(m) in degrees.
- Road/ramp grade: % Grade = m * 100%.
- Vertical lines have undefined slope due to division by zero (Δx = 0).
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