This article is from the Bicycles FAQ, by Mike Iglesias with numerous contributions by others.

From: Jeff Berton <jeff344@voodoo.lerc.nasa.gov>

The grade of an incline is its vertical rise, in feet, per every 100 horizontal

feet traversed. (I say "feet" for clarity; one could use any consistent

length measure.) Or, if you will accept my picture below,

* d | a | o | y R Theta | *___)______________| x

then

Grade = y/x (Multiply by 100 to express as a percentage.)

and

Theta = arctan(y/x)

So a grade of 100% is a 45 degree angle. A cliff has an infinite grade.

[More from Jobst Brandt <jobst.brandt@stanfordalumni.org>]

Date: Mon, 26 Apr 1999 16:11:44 PDT

The steepness of a road is generally measured in % grade, which in

mathematical terms is the slope, or TANGENT of the angle, measured

from the horizontal. This is the ratio of elevation change per

horizontal distance traveled, often called "rise over run". Typically

a road that rises 1-in-10, is otherwise called 10% grade.

Measuring the distance along the surface of the road instead of

horizontally gives practically the same result for most road

gradients. The distance along the road surface gives the SINE of the

angle in contrast to the horizontal distance that gives the TANGENT.

For practical purposes SINE equals TANGENT for small angles (up to ten

degrees or so). For instance, a 20% grade (11.3 degrees), whereas

measuring along the road surface gives a 19.6% grade.

The slope of a road is more useful than its angle because it gives a

direct way to assess the effort required to move forward against the

grade, whereas the angle in degrees does not readily reveal this

information. A 5% grade requires a forward force of approximately 5%

of the vehicle weight (above and beyond the force it takes to travel

similarly on flat ground). A 15% grade requires a propulsion force of

approximately 15% of the vehicle weight.

Although the angle may be more easily visualized, it does not convert

easily to effort without a calculator. For instance a 20% grade is an

11.3 degree angle and is a steep and difficult gradient. The

relationship between angle and slope is non linear becoming 100% (1:1)

at a 45 degree angle. In contrast, the SINE of 45 degrees is 70.7%

while the SINE of 90 degrees (straight up) is 100% for which the slope

(TANGENT) is infinity (or undefined).

The most accurate way to measure this without a precision

inclinometer, is to use a level, a one meter long bar and a metric

ruler. Resting one end of the rod (held level) on the road at a

representative spot, measuring the distance down to the road at the

other end in centimeters gives the percent grade directly. Using a

carpenters level and a one meter long rectangular bar can give

accurate readings to a couple of tenths of a percent.

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