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:

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

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

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