Slope Calculator

Slope is a measure of the inclination or decline of land, road or surface. It indicates the variation in height in relation to a distance traveled.

Slope is generally measured in percentage (a slope of 30%, or a rise/change of 30 units for a run horizontal distance of 100 units).

By extension, some call slope, the angle $ a $ (a slope of 15 degrees).

The slope $ p_\% $ (in % percent) is calculated by the ratio of the height $ h $ by the width $ w $ (multiplied by 100 to get a percentage): $$ p_\% = 100 \times \frac{h}{w} $$

The slope $ p° $ (in ° degrees) is calculated by trigonometric functions: $$ p° = \arctan \left( \frac{h}{w} \right) \times \frac{180}{\pi} $$

The formula for converting slope from degrees to percentage is:$$ p° = \arctan \left( p_\% / 100 \right) \times \frac{180}{\pi} $$

The formula for converting slope from percentage to degrees is: $$ p_\% = 100 \tan( p° \times \frac{\pi}{180} ) $$

A positive slope (or grade) indicates that the land rise or is rising in the chosen direction (positive height difference), while a negative slope indicates that the land is falling or falling in the chosen direction (negative height difference).

A grade/slope of 0% means that there is no significant vertical change in the distance traveled. This is equivalent to flat land (no elevation).

The following equations allow us to find $ d $, $ h $, $ w $ from the angle $ a $ (in degrees)

$$ d = w / \cos(a \pi/180) $$

$$ h = d \sin(a\pi/180) = w \tan(a \pi/180) $$

$$ w = h / \tan(a \pi/180) $$

From the slope $ p_\% $ in percentage:

$$ d = w / \cos(\arctan( p_\% / 100 ) ) $$

$$ h = d \sin( \arctan( p_\% / 100 ) ) = w \tan( \arctan( p_\% / 100 ) ) $$

$$ w = h / \tan( \arctan( p_\% / 100 ) ) $$