Slope Angle

Date: 06/17/2003 at 13:55:13
From: Dee
Subject: Degree, percentage of a slope

What is the difference between the degree of a slope and the 
percentage of a slope?

Date: 06/17/2003 at 16:15:44
From: Doctor Peterson
Subject: Re: Degree, percentage of a slope

Hi, Dee.

People often use misleading terminology in this area. Let me first 
state the proper terms and their definitions, and then come back to 
look at your question.

The SLOPE of a line in geometry, or of a slanted surface such as a 
road or a roof in practical geometry, is the ratio of the "rise" to 
the "run." This might be presented as a fraction or a decimal:

          +              +                      +
        / |             /|                   /  |
      /   |6           / |6               /     |3
    /     |           /  |             /        |
  +-------+          +---+          +-----------+
      6                3                  9

  slope=6/6=1     slope=6/3=2       slope=3/9=1/3

A slope can also be presented as a percentage, which is really just a 
special fraction whose denominator is 100, as indicated by the "%" 
symbol. ("Per cent" means "out of 100.") The three slopes above are 
then 100%, 200%, and 33 1/3 % respectively (obtained by multiplying 
each slope by 100%).

We can also talk about the ANGLE of a slope, which is the angle of 
the slanted line above the horizontal. This is related to the slope 
itself by trigonometry: the slope is the tangent of the angle.

          +              +                      +
        / |             /|                   /  |
      /   |6           / |6               /     |3
    /45   |           /63|             /18      |
  +-------+          +---+          +-----------+
      6                3                  9

  angle=45 deg    angle=63 deg      angle=18 deg

But here is where a lot of people get confused. We say that an angle 
is "45 degrees"; we should NOT say, as many do, that "the degree of 
the angle is 45." That's like saying "the foot of my yard is 100" 
instead of "the width of my yard is 100 feet." The degree is a unit, 
not the name of a quantity.

What confuses things more is that there is another use of the word 
"degree" in English that is more vague, but sounds the same; my 
dictionary defines this as "relative intensity or amount of a quality, 
attribute, or the like." So we could say "what is the degree of the 
slope?" meaning merely "how much slope is there?", without referring 
to the unit or a specific way of measuring the slope.

So when we are talking about numerical measurements, we should not use 
either of the phrases you used, "degree of a slope" or "percentage of 
a slope." We should instead talk about the "angle of the slope" or 
the "percentage slope," or something like that. I can accept "degrees 
of slope," and certainly "a 45-degree slope," because it is clear 
that the unit is being referred to. Likewise, "percentage of slope" 
can be understood (but not "the percentage of a slope," which sounds 
as if you are taking just some part of the slope).

Other terms are used in various fields. In talking about roads, we 
use the terms "grade" or "percent grade"; the angle can then be called 
the "grade angle." The slope of a roof is called the pitch, and is 
usually given as a ratio such as 4 inches per foot or 4 in 12 or 4:12, 
or occasionally as a percentage. Elsewhere, I've seen "gradient" used 
for "slope," and "inclination" for the angle of slope.

A Web search shows many cases where "degree of slope" is used; in some 
it is clear that they are using it to refer to the angle, but in 
others it seems just to mean "amount of slope," and either angle or 
percentage can be used with that term. We probably can't make the 
whole world use the correct terms; but we can try to do so ourselves, 
and when they use misleading expressions we can ask for clarification.

I hope that answers your question. If you have any further questions, 
feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/