# Road Cycling - How do you find % grade of a road?

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cc_rider
06-24-05, 12:37 PM
I was wondering, how you find the % grade of the roads you ride?

Is it published on a bicycle map?
Do you calculate an average using distance and elevation on topo maps?
Do you calculate it using a cyclocomputer and a GPS?
Do you look at the hill and guess? (my current method)
Is there some other device, bike mounted or handheld, for finding the slope of a hill?

I have been wanting to find the grade some of the hills I ride. I have something I use for finding the slope of ramps and roofs on buildings, but I don't want to carry it around, stop, get off the bike mid-climb or mid-descent and measure.

How do YOU find the % grade.

ChAnMaN
06-24-05, 12:42 PM
i dont know this to be a fact but if you have a cyclocomputer with altimiter abilities you might be able use the total distence traveled combined with the vertical distence traveled to make a triangle and then calculate the angle.

just a guess, im sure there are much better ways.

RC2
06-24-05, 12:49 PM
Rise over run... calculate the rise of a hill in ft (or meters, whatever) and divide by it's length. For instance, if you use an altimiter and measure a 100 ft altitude gain over 1000 feet, it's a 10% avg grade. In practice, once you know what 10% looks and feels like, you can subjectively gauge what 5% or 15% are. For reference, I've heard that typical freeway on/off ramps (to an overpass) are graded at 4%...not sure that's true or consistent.

krazyderek
06-24-05, 12:49 PM
i've wondered this aswell, i've thought of going to my local hardware store to see if they have any self leveling angel finder that might do the trick, although that would be off the bike.

Maybe there's some type of gyro built into some computer out there, or i imagine an ALT computer would have the formulas built into it if you read thru the manual

alison_in_oh
06-24-05, 12:49 PM
The accurate way seems to be getting topographic software and making a plot of the elevation change of your road.

There's also cyclecomputers that will do the math of altitude vs. distance for you on the fly. But their altitude meters are calibrated by barometric pressure, so they'll vary to a certain extent.

My method is to pester my most experienced roadie friend who's been on many a road with a marked grade, and to make him estimate the grades of nearby roads. ;)

Brillig
06-24-05, 12:51 PM
How do YOU find the % grade.

Ask that guy in your cycling club who talks about all the grades of all the hills and divide his answer by 2.

;)

06-24-05, 12:51 PM
RC2 nailed it. Rise divided by run. Some cyclo computers with altimeters will give you a grade, but their sketchy at best. You can also use topo map or Dolorme software.

Brillig
06-24-05, 12:53 PM
If it's big enough you can go here http://www.topozone.com/ and calculate rise over run and get fairly close on an average grade. It's really tough to get accurate grades at different points without equipment or plotting the route in topographic software (which isn't always accurate either).

Brillig
06-24-05, 01:11 PM
i dont know this to be a fact but if you have a cyclocomputer with altimiter abilities you might be able use the total distence traveled combined with the vertical distence traveled to make a triangle and then calculate the angle.

Sort of. From there you'd need pythagorus to figure out the "run" part and then you can calculate the grade.

Turn out something like:

gradient = vertical/(Sqrroot(distance^2 - vertical^2))

ChAnMaN
06-24-05, 01:17 PM
Sort of. From there you'd need pythagorus to figure out the "run" part and then you can calculate the grade.

Turn out something like:

gradient = vertical/(Sqrroot(distance^2 - vertical^2))

I was thinking more like using Sin where x is the angle.

Sin(x)=vertical/total....... x= (sin -1) vertical/total

but the rise over run thing is much easier.

TheKillerPenguin
06-24-05, 01:20 PM
For the major climbs, I take a "profile" picture of the grade at one point, using something that sticks straight up as a guide. I then figure out the grade using bitmap, and the rise/run thing. Obviously this only works on the climbs that have a fairly steady climb. For others, I have some topo maps of the area, but they only go down to 100ft isolines so it's hard to be 100% accurate with those. Some others, I just eyeball and use my past experience to guesstimate.

slooney
06-24-05, 01:38 PM
I was wondering, how you find the % grade of the roads you ride?

Is it published on a bicycle map?
Do you calculate an average using distance and elevation on topo maps?
Do you calculate it using a cyclocomputer and a GPS?
Do you look at the hill and guess? (my current method)
Is there some other device, bike mounted or handheld, for finding the slope of a hill?

I have been wanting to find the grade some of the hills I ride. I have something I use for finding the slope of ramps and roofs on buildings, but I don't want to carry it around, stop, get off the bike mid-climb or mid-descent and measure.

How do you find the % grade.

I'm not a mathematician, so don't recall the elegant language for explaining this, but here is my attempt:

First off: Percent slope (grade) = change in elevation in feet/ distance in feet (S=DE/L); 50% slope is one foot travelled vertically for two feet travelled horizontally. 100% slope is 1 foot up for 1 foot travelled.

Degree of slope is often confused with percent of slope. Degree of slope is the measure of the number of degrees in the included angle, measured from level to the top of the hill (90 degrees is the angle when looking straight up, 0 degrees is dead level, 45 degrees is 1 foot up over 1 foot horizontal, or 100% slope). It is not at all uncommon to hear someone say something like "I was on a section of road that was 15 degrees up," which is 26.8 percent slope. Very Steep.

For personal purposes, I take my data from one of a number of mapping programs that give elevation data. These days, I also use readings off a GPS unit (ETrex Legend). Elevation data taken from USGS maps is often not accurate enough to give a good read for specific hill grades- it's very hard to get a good starting and ending elevation from a map that only gives elevation in 20' intervals, as are USGA maps, typically. If your hill is short, accuracy sucks, as your potential for elevation error (as much as 40' off) has a more profound effect on your computation than when the error is incorporated into a big distance. So, accuracy of percent slope estimated from USGA gets better as distance increases.

A clinometer (a hand held device, often integrated into a hand compass, but also available as a stand-alone unit) measures slope, and if used properly (measure from eye height to eye height- not eye height to ground) can be reasonable accurate (for bicylists), and is often used by foresters who need reliable estiation of grade values.

Guessing is rarely accurate for most folks- 2% on pavement looks flat to many. On the other extreme, people often guess a steep section of road as being steeper than it is (a BF poster from a while back was guessing that the slope they were riding up was 30 degrees, or 33% slope- This will never happen on any engineered, paved road in the US, except on the inside of very tight hairpin turns- this is an impossibly steep grade for a road). If you're really good at estimation (because you're a surveyor, civil engineer, landscape architect, or someone else who deals with grade change on a regular basis) you've probably developed a good sense of grade, but this is a skill that takes considerable practice to develop.

Is this any help?

slooney
06-24-05, 01:40 PM
It took me so long to complete my answer that several others beat me to it. Whew boy. :rolleyes:

06-24-05, 02:00 PM
I was thinking more like using Sin where x is the angle.

Sin(x)=vertical/total....... x= (sin -1) vertical/total

but the rise over run thing is much easier.

While I'm enough of a geek to have set up an Excel spreadsheet to calculate these things, just dividing the rise by the distance ridden (the hypotenuse) is pretty close to dividing the rise by the calculated horizontal run for most ridable grades.

Al1943
06-24-05, 02:01 PM
I was thinking more like using Sin where x is the angle.

Sin(x)=vertical/total....... x= (sin -1) vertical/total

but the rise over run thing is much easier.

Your trig equation will give you degrees. Road grades are in % not degrees. A 45 degree angle is equal to 100% grade.

TheKillerPenguin
06-24-05, 02:04 PM
you can find the sides by working backwards through the problem.

sin = O/H Therefore, Hsin = O, or H = O/sin

Edit: with the lengths, you can find the %grade using rise/run.

ChAnMaN
06-24-05, 02:11 PM
Your trig equation will give you degrees. Road grades are in % not degrees. A 45 degree angle is equal to 100% grade.

ohh ok, that makes sense now. I guess I learn something new everyday.

so if you know the angle then can you just divide by 45 to find the grade?

johnny99
06-24-05, 03:55 PM
ohh ok, that makes sense now. I guess I learn something new everyday.

so if you know the angle then can you just divide by 45 to find the grade?

tangent(angle) * 100 = grade
tangent is not linear

ChAnMaN
06-24-05, 04:24 PM
tangent(angle) * 100 = grade
tangent is not linear

what am i doing wrong?

Tan(45)*100=161.977

i thought a 45 degree angle was 100 % grade

johnny99
06-24-05, 04:35 PM
what am i doing wrong?
Tan(45)*100=161.977
i thought a 45 degree angle was 100 % grade

tangent(45 degrees) = 1.0
You probably forgot the difference between degrees and radians.

zonatandem
06-24-05, 04:48 PM
There is a bicycle device available called and 'inclinometer'.

zonatandem
06-24-05, 05:01 PM
Inclinometer info:
Info@Velimpex.com
Device clamps onto handlebar. Weighs 28 grams (1 ounce) and costs about \$25. Will measure gradient -21 % to +21%.

mnutini
06-24-05, 05:13 PM
There is a bicycle device available called and 'inclinometer'.

You can build your own.

or do it by using a level

va_cyclist
06-24-05, 06:02 PM
Not to be geometrically anal about it, but does measuring road distance give you an accurate rise-over-run? Specifically, the run part? If your hill forms a right triangle, then the short leg of the triangle is your rise, that's straightforward; but in a true slope calculation, the "run" is the bottom leg of the triangle, not the sloped leg (hypotenuse), which is what's being measured if you rely on your cyclocomputer's mileage. The difference is probably trivial, but might cause you to slightly underestimate the slope (too long a run for a given rise). Any math weenies have insight into this?

johnny99
06-24-05, 06:08 PM
Not to be geometrically anal about it, but does measuring road distance give you an accurate rise-over-run? Specifically, the run part? If your hill forms a right triangle, then the short leg of the triangle is your rise, that's straightforward; but in a true slope calculation, the "run" is the bottom leg of the triangle, not the sloped leg (hypotenuse), which is what's being measured if you rely on your cyclocomputer's mileage. The difference is probably trivial, but might cause you to slightly underestimate the slope (too long a run for a given rise). Any math weenies have insight into this?

If the grade is less than 20% or so, the difference between horizontal and hypotenuse is probably significantly less than your measurement error. For very short steep hills, you can convert using trigonometry or pythagoras.

va_cyclist
06-24-05, 06:13 PM
Re: the inclinometer...there have been several posts by folks who found them worthless because of their sensitivity to changes in momentum; i.e., pedaling motion and uphill acceleration cause the inclinometer to swing around and give poor readings. They seem to work fine if you come to a complete stop on a hill.

Brillig
06-24-05, 06:38 PM
Not to be geometrically anal about it, but does measuring road distance give you an accurate rise-over-run?

No, thus the corrections I laid out in post 9. Technically, knowing the hypotenuse of a triangle (the road distance) and the "rise" you can figure out the "run" and then calculate rise over run from that.

krazyderek
06-24-05, 06:51 PM
You can build your own.

thanks, i've been trying to find an alternative to running around the city with a 3 foot level :)

krazyderek
06-25-05, 11:53 AM
took 10 mins to make, just lined up the outer verical line with the side of the case and taped it, then drilled a little hole for the thread. Dead on when i calibrated with a level.

Thanks again :)
(this is laying on my scanner that's why it's not zero'd)

JavaMan
06-25-05, 12:08 PM
This is the correct formula. RISE is in feet, DISTANCE is in miles.

Put it into a spreadsheet for future use.

ke422azn
06-25-05, 12:28 PM
what a dumb question, this guy never taken geometry or something? Stop biking so much, go take some math classes

Wurm
06-25-05, 12:36 PM
What does it really matter what the exact gradient is - 5%, 8%, 10%, 14% etc? To me, a climb is:

1. Pretty easy
2. Sorta hard
3. Really hard
4. A @!#&ing wall

Big deal! :rolleyes:

Longhorn
06-25-05, 05:52 PM
I just got one of these to use with my trig students but I also plan to use it to check out the grade of some of my hills.

http://www.etacuisenaire.com/catalog/product?deptId=FLIGHTBUBBLES&prodId=8713&cs=9000

ke422azn
06-25-05, 05:55 PM
dont most people know angles and similar triangles before they go into to trigonometry?

ChAnMaN
06-25-05, 07:32 PM
dont most people know angles and similar triangles before they go into to trigonometry?

yes but find the the grade isnt all just common sense. I did all my math right and i came out with a degree answer, i still had to be informed that the grade is a precentage of a 45 degree angle. Something you dont learn in geometry.

johnny99
06-25-05, 07:37 PM
yes but find the the grade isnt all just common sense. I did all my math right and i came out with a degree answer, i still had to be informed that the grade is a precentage of a 45 degree angle. Something you dont learn in geometry.

Grade is not a percentage of a 45 degree angle. Grade is the tangent of the angle, aka rise/run. You probably learned all of this in high school, though the names may be different. Grade or gradient is more of an engineering term than a pure math term.

krazyderek
06-27-05, 07:41 PM
Ok i took some measurements with my little cd case inclinometer, just want to make sure the scale is right.

Can one of you math wizze's tell me how high to adjust my inclinometer if the base (run) is 121mm (4.763") long so as to get a 20% grade reading? I think the proper rise is 23mm (0.905")??

If so then this thing is correct, and my local "big hills" aren't very big after all :(

EDIT: Ok, figured out how to use excel with one of the simpler ( the first) formula, seems the inclinometer is right on :)

Al1943
06-27-05, 08:26 PM
DeLorme sells a set of CD's with topographic maps of the entire U.S. (Topo USA) at a very reasonable price. With these it is very easy to plot a road on a hill, DeLorme's software will give you a profile with the rise, run, and grade in %.

Al

johnny99
06-27-05, 08:35 PM
DeLorme sells a set of CD's with topographic maps of the entire U.S. (Topo USA) at a very reasonable price. With these it is very easy to plot a road on a hill, DeLorme's software will give you a profile with the rise, run, and grade in %.

Al

DeLorme's grade feature is very inaccurate, especially for long climbs with switchbacks. They are often large errors in both the total distance and the total elevation gain.

ExMachina
06-27-05, 09:47 PM
Ok i took some measurements with my little cd case inclinometer, just want to make sure the scale is right.

Can one of you math wizze's tell me how high to adjust my inclinometer if the base (run) is 121mm (4.763") long so as to get a 20% grade reading? I think the proper rise is 23mm (0.905")??

First, the base of your inclinometer is not the run, it's the hypotenuse

Not sure what formula you are using but the correct solution to your question is:

=SIN(ATAN(.20))*121 or
=SIN(ATAN(.20))*4.769

Therefore, the rise for your inclinometer on a 20% grade should be 23.7mm or 0.934"

krazyderek
06-27-05, 09:59 PM
First, the base of your inclinometer is not the run, it's the hypotenuse

Not sure what formula you are using but the correct solution to your question is:

=SIN(ATAN(.20))*121 or
=SIN(ATAN(.20))*4.769

Therefore, the rise for your inclinometer on a 20% grade should be 23.7mm or 0.934"

Thanks for double checking for me :) Numbers have a way of tricking me sometimes :p

tanguy frame
06-28-05, 12:42 AM
Unitl it quit on me, I used a Vetta V100A which has a %grade feature. It was consistently overestimating actual grade by 2X, but I got used to what it called a 5%, 12% 15% (22%!) etc. grade. To get approximate grade, I divided its readings by 2. now I have a feel for it, and I can sort of guess. Who cares anyway? The thing broke, and I'm much less distracted now.

Don Cook
06-28-05, 08:15 AM
There have been some interesting responses to this. The most direct would be to measure the distance in linear feet traveled and the difference in altitude between your starting point and your ending point. Your odometer function can provide you you linear feet, but then you'd nee a rather sensitive altimeter to get the second measurement. Another approach is to used the Euclidian geometric Angle-Side-Angle or Angle-Angle-Side functions. What you need to make the calculation this way is:
1. Standing at the bottom of the incline use whatever method available to you to estimate the angular degree of the slope. If you have access to an inclinometer it makes things a bit easier and more accurate. You could improvise one by using a school childs compass and a piece of string (nylon fishing line might be better). Prepare the compass by turning it curved side down and directly across from the 90 degree point fix the string. Weight the string with a small nut. When the compass is held upside down, and level, the string should be hanging straight down and in line with the 90 degree mark. Now if you sight along the flat edge of the compass and raise the angle of the compass so that it is pointing directly at the summit of the inlcine, have a friend note the number of degrees that the weighted string has changed from the 90 degree mark. That is the approximate angle of the incline.
2. Measure the linear distance from the bottom up the inclined plane. Use anything with an odometer and just convert to feet.
3. Now the change in altitude can be determined by the A-A-S geometry.
In the equation altitude is 'c' (this is what we've got to calulate)
The measured incline will be 'C'
The distance you measured up the slope is 'a'
And the base angle is 'A' (which is 90-C)
(a)* sin (A)
c = -------------
sin (C)

After calculating 'c' simply divide it by 'a' to get the gradient. Multiply the result by 100 to convert it to the more familiar percentage.

Example: Suppose you approximate a 5 degree slope using whatever method you choose. You measure the linnear feet to the top, it's 380ft.

The equation is 380 * sin A (A=85, that's 90 minus the 5 degerees measured)
divided by sin (C). So we have

380 * .17608
------------ = 66.771 (thats 66.771 feet of altitude change)
.95898

66.77 divided by the 380 linear feet of the slope gives a gradient of 18%

ExMachina
06-28-05, 11:01 AM
There have been some interesting responses to this. The most direct would be to measure the distance in linear feet traveled and the difference in altitude between your starting point and your ending point. Your odometer function can provide you you linear feet, but then you'd nee a rather sensitive altimeter to get the second measurement. Another approach is to used the Euclidian geometric Angle-Side-Angle or Angle-Angle-Side functions. What you need to make the calculation this way is:
1. Standing at the bottom of the incline use whatever method available to you to estimate the angular degree of the slope. If you have access to an inclinometer it makes things a bit easier and more accurate. You could improvise one by using a school childs compass and a piece of string (nylon fishing line might be better). Prepare the compass by turning it curved side down and directly across from the 90 degree point fix the string. Weight the string with a small nut. When the compass is held upside down, and level, the string should be hanging straight down and in line with the 90 degree mark. Now if you sight along the flat edge of the compass and raise the angle of the compass so that it is pointing directly at the summit of the inlcine, have a friend note the number of degrees that the weighted string has changed from the 90 degree mark. That is the approximate angle of the incline.
2. Measure the linear distance from the bottom up the inclined plane. Use anything with an odometer and just convert to feet.
3. Now the change in altitude can be determined by the A-A-S geometry.
In the equation altitude is 'c' (this is what we've got to calulate)
The measured incline will be 'C'
The distance you measured up the slope is 'a'
And the base angle is 'A' (which is 90-C)
(a)* sin (A)
c = -------------
sin (C)

After calculating 'c' simply divide it by 'a' to get the gradient. Multiply the result by 100 to convert it to the more familiar percentage.

Example: Suppose you approximate a 5 degree slope using whatever method you choose. You measure the linnear feet to the top, it's 380ft.

The equation is 380 * sin A (A=85, that's 90 minus the 5 degerees measured)
divided by sin (C). So we have

380 * .17608
------------ = 66.771 (thats 66.771 feet of altitude change)
.95898

66.77 divided by the 380 linear feet of the slope gives a gradient of 18%

If you know the angle you know % grade...am I missing your point?

ChAnMaN
06-28-05, 11:15 AM
ok im confused, so if i have an angle of 21 degrees what is the gradient?

Don Cook
06-28-05, 11:15 AM
If you know the angle you know % grade...am I missing your point?

You would have been correct if you'd stated that "when we know the angle you can determine the %grade". When we describe a grade in percentage terms, it is a function of the height of a slope divided by the length of the slope. If you will note in the example of mine that you replied to, the slope (or angle) was 5 degrees, yet the percent grade was 18%. My post was to offer a practical way to determine the percent grade when you don't have the means to measure the height of the slope directly. Not everyone has access to an altimeter with accuracy to a few feet. It is true that as the angle of the slope increases in degrees, the resulting "percentage grade" would also increase, but they're not the same arithmetically.

ExMachina
06-28-05, 11:23 AM
You would have been correct if you'd stated that "when we know the angle you can determine the %grade". When we describe a grade in percentage terms, it is a function of the height of a slope divided by the length of the slope. If you will note in the example of mine that you replied to, the slope (or angle) was 5 degrees, yet the percent grade was 18%. My post was to offer a practical way to determine the percent grade when you don't have the means to measure the height of the slope directly. Not everyone has access to an altimeter with accuracy to a few feet. It is true that as the angle of the slope increases in degrees, the resulting "percentage grade" would also increase, but they're not the same arithmetically.

A 5-degree slope is an 8.7% grade (tan(5 degrees)) no matter how long it is, just like a 45 degree slope is a 100% grade regardless of length...

ExMachina
06-28-05, 11:24 AM
ok im confused, so if i have an angle of 21 degrees what is the gradient?

That's a 38% grade... tan(21 degrees)*100

johnny99
06-28-05, 11:25 AM
ok im confused, so if i have an angle of 21 degrees what is the gradient?

tangent(21 degrees) * 100 = 38% grade
Where can I find this road?

ChAnMaN
06-28-05, 11:40 AM
tangent(21 degrees) * 100 = 38% grade
Where can I find this road?

i just made up a number so someone would just do a problem for me, and now i get it. i was just entering in 21 not 21 degrees. i think i get it now.