I’ve seen the grade of a road generally described as the “rise over the run times 100”. Basically, this is the definition we learned in elementary algebra except it is expressed as a percentage. So the grade of a road that has a 2 degree slope would be 100*TAN (2 degrees) = 3.49%
Accorcing to this calculation, a road that has a 30 degree angle of tilt would have a grade of 57.7% and
a road that has a 45 degree angle of tilt would have a grade of 100%.
Sometimes though, I have read about the grade of a road being calculated by the rise divided by the distance traveled which would be the hypotenuse of a triangle.
So, by this method, a 30 degree angle would be considered a grade of 100*SIN(30 degrees) = 50% and
a road that has a 45 degree angle of tilt would have a grade of 70.7%.
So, I know how to calculate the grade of a road by both methods but who uses the second method? Yes, for small angles (the grades of almost all roads), the difference between the 2 methods is very small. However when angles get above 10 degrees, the differences become significant. So, who would use this second type of measurement? Would it be someone who isn’t measuring grades of roads at all such as carpenters, roofers, mountain climbers?
Civil engineer here. I’ve never heard the grade of the road defined as anything but the rise over the run expressed as a percentage.
Your second method seems to be simply an approximation of the first, since in the field it’s easier to measure the hypotenuse than the true horizontal distance.
As you state, this does not produce much of a difference for small slopes, but starts to break down for larger ones.
In summary, then, the first method is correct. The second method is incorrect, but works OK as an approximation for small slopes (like those typically found for roads).
That’s interesting. I always naively thought that 90[sup]o[/sup] was considered a 100% slope.
If 45[sup]o[/sup] is considered 100%, then you could have slopes of greater than 100%, in theory. Even though you would never build a road like that on purpose, you might take an SUV up a hillside of that slope.
Damn. Scooped by robby yet again. Always a bridesmaid…
Okay, let’s find something to add here. When designing a roadway, a percentage is used to define the profile (longitudinal section) of the road. The cross pitch (cross section) is usually also defined as a percentage (say 2%), but is occasionally referred to as inches/foot (say 1/4"/ft, which is close enough to 2% as to make no difference).
A similar thing happens with piping. When our plumbing engineer is running gravity pipes (sewer or drain) inside the building, he uses inches/foot. When I pick them up outside, I use percentages.
This is purely a guess, but as far as I can determine the designation generally follows the scale of the drawing. Plumbing drawings are drawn in architectural scale (say 1/4"=1’-0"), and use inches/foot for slopes. Site plans use engineering scale (1"=20’) and use percentages. However, site details such as cross sections are typically drawn using architectural scale, which may explain the residual use of inches/foot.
When expressing the grading of slopes, we get into an odd situation. You’re probably familiar with roof pitches expressed as rise/run (say a 4/12 roof, which rises 4 inches in 12 inches). Well, for some reason slope grading reverses those, so a 3:1 slope rises 1 foot in 3 feet). Go figure.
So, using the examples above, the following are all equal:
Steep and/or long gradients on roads in the UK used to be signposted with a ratio - 1:10 (1 in 10 - you go up 1 for every 10 units you drive forward, along the hypotenuse, I think). The convention for the modern signage is the percentage gradient, but plenty of examples of the old signs still survive (or at least they did last time I did any large amount of driving in the west country, a couple of years back).
Yes, I always thought the rise to run ratio was the one that always should be used.
However, I’ve seen on other message boards (maybe not the most reliable of sources) that for example, if you are climbing a hill or mountain, it would be misleading to say 45° is a 100% slope (using the tangent formula). To the average person, a 100% slope sounds like the wall of a cliff.
As the angle gets larger, the slope by tangent formula seems even more confusing. For example for a 75° angle, the grade by the sine function is 96.59% but by the tangent function it is 373.21%. Also, wikipedia gives a quick reference (and table) to grade calculated by sine in an article here. The majority of the article deals with grade calculated by the tangent formula.
Maybe a hijack but . . .
There is such a thing as a gradian of which there are 100 in a right angle. Modern calculators rarely have them but they were once common during the 80’s (the famous D-R-G button on the Casios). Being the math geek I am, I always thought this was how they measured hill slopes as a percent.
Gradians (or grads) are not slope percentages. I’ve seen them used exclusively in French ground survey plans, and even then only in limited application. They are, as describe, a decimal alternative to degrees (and a painful one to use).
The slope is the rise over the run, the same as the tangent of the angle of the slope. Yes, this means that a 45° angle has a 100% slope (tan 45° = 1.000), and a 60° angle has a 173% slope (tan 60° = 1.732). A vertical wall would have infinite slope. This isn’t a problem for anyone except those who insist that there cannot be anything more than 100% of a given measure.
It’s not a problem, it’s just counterintuitive that you would have greater than a 100% slope. I would love to get a 110% raise, have an IQ that is 130% of average, or get a 300% return on an investment. But I can’t be 105% healthy, lose 125% of my body fat, or have a system with 115% availability, and it is counterintuitive to have more than a 100% slope. I can understand it mathematically but it’s not intuitive.
Here’s a website (Riverside County California) that states “Slope measures variation from the horizontal. A flat terrain is 0% and and a vertical cliff is a 100% slope.”
Well, I wrote a calculator for this and it is located here
It computes slopes grades and gradients by both the tangent and sine functions.
It agrees with the table in the Wikipedia article here
Just by looking at the comments in Wikipedia, the article may not be very well written. For one thing the article states there are 3 “systems” for indicating a highway’s slope 1) the angle 2) a percentage (such as a 2% grade) and 3) using the sine function. If anything a third system would be a ratio such as 1 in 20. This being the case, these 3 “systems” could be used for the tangent function and the sine function.