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# How to Calculate % Grade?

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Road Cycling “It is by riding a bicycle that you learn the contours of a country best, since you have to sweat up the hills and coast down them. Thus you remember them as they actually are, while in a motor car only a high hill impresses you, and you have no such accurate remembrance of country you have driven through as you gain by riding a bicycle.” -- Ernest Hemingway

# How to Calculate % Grade?

06-26-06, 09:11 AM
#1
markwebb
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How to Calculate % Grade?

How do you calculate % grade. As an example, 100 foot rise over 1,320 feet (0.25 miles) = x% grade. How do you calculate it?
06-26-06, 09:15 AM
#2
Alphamoose
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That's 7.5% (100/1320)*100.
06-26-06, 09:19 AM
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kyledr
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So this means a 1000 foot rise over 1000 feet is 50% grade and 45 degrees, for instance. So if you want to convert from % to degrees multiply the percent in decimal form (multiply by .01 I think) by 90.
06-26-06, 09:20 AM
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TheKillerPenguin
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How high the climb is in ft divided by the distance climbed in ft. Multiply the answer by 100.
06-26-06, 09:33 AM
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KevinF
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Originally Posted by kyledr
So this means a 1000 foot rise over 1000 feet is 50% grade and 45 degrees, for instance.
This is where it gets confusing (IMHO). Percent grade calculations are done by the "rise over run" formula (vertical elevation change divided by distance travelled). Typical roads aren't all that steep, so the distance ridden (i.e., the hypotenuse) is essentially the same as the true "run" of the right-triangle being formed by the hill-climb. So you can use the rise / distance-ridden formula to get pretty close to the true grade of the road.

However, in your example of a 1000 foot rise over 1000 feet the hypotenuse and the "true run" are no longer close. The way your statement is worded could mean that you have either a 45 degree angle (if your 1000 feet references the "true run") or you could have a 90 degree angle -- the only way you climb 1000 feet in 1000 feet of riding is if you literally rode straight up.
06-26-06, 09:36 AM
#6
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What does percent grade mean on a road?

When you travel through the mountains, you often see signs that say things like "Trucks check brakes -- 10% grade" or "6% grade -- Trucks use right lane only." These numbers obviously have something to do with the steepness of the road, but their exact meaning is a mystery to most drivers.
The grade of something is simply a measure of its rise over its run. To understand rise and run, it helps to think of the hill as a big right triangle (a triangle with a 90-degree angle), like this:

The rise is the length of side B, or the height of the hill. The run is the length of side A, the horizontal measure of the hill at ground level. So, if you rose 100 feet over a horizontal distance of 1,000 feet, rise over run would equal 100 divided by 1,000, or 0.1. To get the percent grade, you simply multiply by 100, which gives you 10%. It doesn't matter whether you use feet, meters, miles or kilometers -- if you know how far the road rises in a given horizontal distance, you can calculate the percent grade.
In common practice, people often refer to percent grade as the rise divided by the distance you would travel going up the hill (side C), rather than the horizontal distance (side A). If you have an odometer and an altimeter, this is pretty easy to calculate. You check the altitude at the starting point and reset the odometer trip meter. You climb a certain distance and divide the change in altitude by the miles you've traveled. This is not technically the grade, but for normal roads that aren't very steep, it ends up being pretty close because the horizontal distance and the length of the actual road are nearly the same.
To calculate percent grade exactly, you need to figure out the horizontal distance traveled (A). Since you know B and C, you can calculate A using the Pythagorean theorem -- "the square of the hypotenuse is equal to the sum of the squares of the other two sides," or:
C2 = A2+ B2
This means that:
A = SquareRoot (C2 - B2)
If you had driven 1,000 miles down the road and risen 100 miles, the horizontal distance would be the square root of 990,000, which is approximately 994.99 (rounded up).
So what good does all of this do you? First of all, the percent grade gives you a relative sense of how steep the hill is. If you've climbed a hill designated as having a 5% grade, for example, you'll know approximately what to expect from any other 5% grade hill.
. Percent grade also gives you enough information to figure out the grade angle -- the angle of the hill's ascent (angle d in the diagram below). If you know trigonometry, you've probably recognized that rise divided by run is equal to the tangent of angle d. To calculate the angle of ascent when you know the percent grade, you simply take the arc tangent of the grade (the inverse of the tangent). So, if you have a 10% grade, you look up the arc tangent of 0.1 and find that the angle is 5.71 degrees.

Roads are measured in percent grade rather than degrees for two reasons:
· You don't have to have a special calculator or trig tables to calculate percent grade.
· If you know the percent grade, it's easy to calculate the approximate distance you have risen or fallen simply by looking at your odometer. If you have coasted down a mile-long hill on a 10% grade, you know you have fallen about a tenth of a mile in vertical height.
06-26-06, 10:14 AM
#7
NoRacer
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Percent slope =

%m = (rise/run)*100

%m = (100/1320)*100
%m = (.0757)*100
%m = 7.6
09-11-11, 12:13 PM
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quickquest88
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Originally Posted by kyledr
So this means a 1000 foot rise over 1000 feet is 50% grade and 45 degrees, for instance. So if you want to convert from % to degrees multiply the percent in decimal form (multiply by .01 I think) by 90.
sorry to revive a dead thread, but i just looked this up today after seeing someone on tv clearly misunderstand this.
in this particular example
(rise/run)*100
(1000/1000)*100
(1)*100
100

a 45 degree hill is clearly a 100 grade. as a previous poster mentioned, some people miscalculate this by using sides C and B instead of the proper sides A and B, as refrenced by the pythagorean theorum. a 50% grade is 22.5 degrees.10% grade is 4.5 degrees.
09-11-11, 12:19 PM
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ilovecycling
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09-11-11, 01:20 PM
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Originally Posted by Alphamoose
That's 7.5% (100/1320)*100.
This (post #2) should have ended the thread. Over 5 years ago.
09-11-11, 06:30 PM
#11
bobbycorno
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Originally Posted by kyledr
So this means a 1000 foot rise over 1000 feet is 50% grade and 45 degrees, for instance. So if you want to convert from % to degrees multiply the percent in decimal form (multiply by .01 I think) by 90.
No. 1000' in 1000' = 100%. Zero percent is perfectly flat, and straight up is infinite percent (no matter how far you climb, you never go forward, so you'd be dividing by zero, which is by definition infinity.

SP
Bend, OR
09-11-11, 07:05 PM
#12
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Originally Posted by ilovecycling
IDK, I tell people my Garmin readings on here, and they call me a liar. Apparently there are no roads that peak out at 28% in the rest of the world.
09-11-11, 08:23 PM
#13
nhluhr
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Most people mistakenly use distance traveled as their denominator. That works only with some trigonometry. The correct "run" is the horizontal distance traveled.

Still, for relatively low grades, it's close enough for bragging rights.
09-11-11, 08:43 PM
#14
canam73
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For cycling purposes this really was answered with the first response 5 years ago. Even taking Soloist's extreme example of a 28% grade as an example the change between using the the hypotenuse and the flat leg as your denominator changes the result by a little over 1%. At 28% it won't help you. And for more common (i.e. lesser) pitches the difference will be less. At 10% the difference is .05%

Hey, let's have the math dorks really whip it out by calculating pi to the nth decimal, where n=stupid.
10-25-13, 08:04 PM
#15
kocoman
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Can I use the theodolite app to calculate it? thanks
10-25-13, 09:17 PM
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Holy resurrected thread BatMan!
10-26-13, 04:09 AM
#17
Pirkaus
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Zombies

Originally Posted by MikeyBoyAz
Holy resurrected thread BatMan!
Just in time for Holloween

Pirk
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10-26-13, 05:24 AM
#18
Cyclelogikal
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Rise over run................and multiply by 100

So 100/1320x100= 7.5%

Last edited by Cyclelogikal; 10-26-13 at 05:28 AM.
04-05-14, 10:36 PM
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I looked this up on wikipedia - and the popular answer here is incorrect

Wikipedia: Grade (slope) - Wikipedia, the free encyclopedia

Pertinent section:

as a percentage, the formula for which is 100 \frac{\text{rise****{\text{run**** which could also be expressed as the tangent of the angle of inclination times 100. In the U.S., this percentage "grade" is the most commonly used unit for communicating

So for the given case the answer is 100/(1320^2-100^2)^(1/2) ~= 7.60 degrees

However, the back-of-envelope calculation of simply taking rise/run diverges by a factor of < 0.005 for angles under 10% and exceeds 0.01 only for grades > 20%

E.g. 100/1320 is 7.57 - compared with the precise calculation of 7.60, an inaccuracy of just 0.28%

Last edited by javadba; 04-05-14 at 10:53 PM.
04-05-14, 11:10 PM
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04-06-14, 12:17 AM
#21
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Originally Posted by ilovecycling
Except if you buy one that doesn't display grade.
04-06-14, 07:43 AM
#22
gc3
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Originally Posted by javadba
I looked this up on wikipedia - and the popular answer here is incorrect

Wikipedia: Grade (slope) - Wikipedia, the free encyclopedia
that's because wikipedia didn't exist when the thread originated...
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04-06-14, 11:14 AM
#23
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Just to beat this ancient dead horse a little more, AKAIK, GPS algorithms don't actually give you the distance traveled on a slope, but rather the distance on a map between two waypoints, not accounting for grade. This means that for hilly terrain, they actually underestimate distance traveled on the road by a very small amount. But it also means that they give you the "run" in rise over run, without needing to convert from the hypotenuse distance. OTOH, if your distance comes from a sensor on your wheel, you do have to convert.

As a general rule, this consideration as well as pretty much all the rest in this thread, are less important than the fact that the measurements available (precise run, precise rise) are seldom accurate enough for the corrections to matter (particularly for grades less than, you know, 20% or so.
03-27-17, 07:05 PM
#24
Crazyraceguy
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No it's not.

Originally Posted by bobbycorno
No. 1000' in 1000' = 100%. Zero percent is perfectly flat, and straight up is infinite percent (no matter how far you climb, you never go forward, so you'd be dividing by zero, which is by definition infinity.

Bend, OR
100% grade is when the run and rise are equal.
That would be a 45º angle. Beyond the the % grows exponentially to the point that at 84.6º you have 1000% grade and 90º is litterally infinity because you only travel up and not foreward at all.
03-27-17, 07:48 PM
#25
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