The Slope Formula
Slope (m) measures the steepness of a line and is defined as rise over run: m = (y₂ − y₁) / (x₂ − x₁). For two points (1, 2) and (4, 8): m = (8 − 2) / (4 − 1) = 6/3 = 2. For every 1 unit increase in x, y increases by 2.
Slope values and their meaning: Positive slope = line rises from left to right. Negative slope = line falls from left to right. Zero slope = horizontal line. Undefined slope = vertical line.
Slope angle: tan(θ) = slope. For a slope of 1: θ = arctan(1) = 45°. For a slope of 0.5: θ = arctan(0.5) = 26.6°. Engineers and architects often work in both slope ratios and degrees depending on context.
Slope in the Real World: Roads, Ramps, and Roofs
Slope has direct practical applications across engineering and construction.
Road grades: Road steepness is expressed as a percentage grade (rise/run × 100). A 6% grade means 6 feet of elevation gain per 100 feet of horizontal distance. The steepest sustained grade on a US Interstate highway is around 6–7%. Residential streets rarely exceed 15–20%. The steepest road in San Francisco (Filbert Street) has a grade of 31.5%.
ADA wheelchair ramp requirements: The Americans with Disabilities Act mandates a maximum slope of 1:12 (1 inch of rise per 12 inches of run = 8.33% grade) for wheelchair ramps. For a 24-inch platform height, the ramp must be at least 24 feet long. This is why building accessibility often requires long ramps or switchbacks for significant height changes.
Roof pitches: Expressed as rise/run in inches per foot. A 4/12 pitch means 4 inches of rise per 12 inches of run = 18.4° angle. A 6/12 pitch = 26.6°. A 12/12 pitch = 45°. Steeper roofs shed snow better and allow more attic space but require more material and are harder to work on.