# Calculating Percentage of Slope

#### meppwc

I am looking for a formula to determine percentage of slope of a hill.
For example

If I stretch a string for a horizontal distance of 10 feet
And the vertical drop is 5 feet at the end of the string

What percentage slope do I end up with?

Can someone provide the formula to calculate this?

#### ebaines

The slope as a percentage is calculated from slope = 100 x rise/run, where "run" is the horizontal distance covered. This is equivalent to saying that slop= tangent of the angle to the horizontal expressed as a percentage. In your example the slope would be 100 x 5/10 = 50%. A slope of 100% is equivalent to a 45 degrees, and slope greater than 100% means the angle is greater than 45 degrees.

In some cases you may need to calculate the "run" distance using the Pythagorean theorem. If the length of the string in your example is known to be 10, and the drop is known to be 5, then the horizontal run is:

run=sqrt(10^2-5^2) = 8.66,

and the slope is then 100 x 5/8.66 = 57.7%. For "reasonable" values of slope such as engineers might use for calculating slope on a roadway (typically less than 10%) it's usually "good enough" to use slope = rise/string length and not bother calculating horizontal run.

Last edited:

#### Plato

MHF Helper
meppwc;885586 If I stretch a string for a horizontal distance of 10 feet And the vertical drop is 5 feet at the end of the string What percentage slope do I end up with? Can someone provide the formula to calculate this?[/QUOTE said:
First, I have never seen the phrase percentage slope before.
You may want to give a detailed definition.

Mathematically it seems that you have a right triangle. The legs are $10~\&~5$ so the the hypotenuse(length of driveway) is $\sqrt{125}$. The angle of depression is $\arctan(-0.5)=-26.57 degrees$.

Those are the calculations, but I don't know what "percentage slope" means.

#### meppwc

Maybe I should have been more accurate with what I am trying to do.
I was told by a tool rental company that it is unsafe to use their hoist if the slope of my ground is > 12.5%

So I ran the string 10 feet (level). And then measured from the end of the string to the ground and came up with 5 feet.
So based on that I am trying to find out if I am under/over the 12.5% limitation.

#### ebaines

Yes, you are over the limit. A 12.5% slope means that over ten feet of horizontal distance you would have a maximum rise of 10 x 0.125 = 1.25 feet. Your rise of 5 feet means you have a 50% slope (as noted earlier), which is much steeper than the allowed maximum.