steepness

Steepness is the degree to which a surface rises over a horizontal distance. It is a fundamental geometric property of slopes and is used in geography, engineering, construction, and everyday description. Steepness can be described in several equivalent ways, such as a gradient, a percent grade, or an angle of incline.

The gradient measures rise over run (slope = vertical change divided by horizontal distance). The same value is often expressed as a percent grade, calculated as rise divided by run times 100. The angle form relates to gradient by tan(theta) = gradient, so theta = arctan(gradient). For practical conversion: 5% ≈ 2.9°, 10% ≈ 5.7°, 20% ≈ 11.3°.

Measurement and calculation methods include clinometers, theodolites, digital terrain models, LiDAR, and GPS-based devices to estimate steepness. In geology and geomorphology, slope distributions are characterized from elevation data; in civil engineering, maximum designed grades are specified for roads and railways based on vehicle performance and safety considerations.

Applications and implications of steepness span roads, railways, ski slopes, and building on sloped terrain. Steepness influences drainage, stability, accessibility, and safety for hikers and climbers. Perceived steepness can vary with surface texture and context, making precise communication about slope important in planning and design.