Slope and Grade from Contour Lines

ScopeCount the contour lines the route crosses and multiply by the interval for the rise; scale the measured map distance for the run; everything else is one arctangent. Four different trades ask for the answer in four different units -- a grading contractor wants percent, a hiker or an avalanche forecaster wants degrees, an earthwork crew wants the ratio, and anyone estimating time or rope wants the true slope distance rather than the map distance -- so all four are printed together. The relationship between percent and degrees is worth having in front of you because the two scales diverge sharply: 100% grade is 45 degrees, not 90, and the 30 degrees that matters most to an avalanche forecaster is a 58% grade. Reading one scale as if it were the other is a genuine safety error in steep terrain. Five 40 ft intervals over a map distance that scales to 1,300 ft is a 200 ft rise, a 15.4% grade, 8.75 degrees, 6.5 to 1, and a slope distance of 1,315 ft. Note how little the slope distance exceeds the map distance -- fifteen feet out of thirteen hundred, about 1% -- and that stays true up to surprisingly steep ground, since even a 30% grade is only 4.4% longer than the map. It is elevation gain, not path length, that makes steep ground slow. A map reading; the ground itself, the map's own contour accuracy, and the party's judgment govern.

Formula or tablerise_ft = contour_interval_ft x intervals_crossed; run_ft = map_distance_in x representative_fraction / 12; grade_pct = rise/run x 100; slope_angle_deg = atan(rise/run); slope_ratio = run/rise; slope_distance_ft = sqrt(run^2 + rise^2).

Source and editionSlope from contour count and map scale, reported as percent, degrees, ratio, and true slope distance, by name; public map arithmetic and trigonometry. The ground itself, the map's contour accuracy, and the party's judgment govern.

Free accessCounting contours and scaling a distance are public map arithmetic; the arctangent is trigonometry.

What governsEstimate. AHJ and licensed professional govern.

API field namescontour_interval_ft, intervals_crossed, map_distance_in, representative_fraction, rise_ft, run_ft, grade_pct, slope_angle_deg, slope_distance_ft

• Rise from contours interval times the number of intervals crossed; the map's contour accuracy governspublic map arithmetic
• Percent is not degrees 100% grade is 45 degrees; 30 degrees is a 58% gradetrigonometry
• Path length is not the cost slope distance exceeds map distance by only a few percent even on steep groundtrigonometry