Slipstream - June 2020

16 May
Nerd Alert: What Does “Percent Grade” Really Mean?
by Carey Spreen, Certified Nerd
At some point in your life you may have wondered what those
“6% Grade” or “8% Grade” signs you see on steeply inclined
roads really mean, and how they correlate to the angle of the slope
in degrees – I know I have! If you haven’t ever wondered that, you
may just want to stop reading this right now. Math ahead!
Thanks to those of you who are still reading, thank you – I
hope you’ll find it interesting.
About a year ago I signed up for one of the 90-minute sessions
at the Porsche Experience Center in Atlanta, where you can drive a
model of your choice (and sometimes two models of your choice)
to get a feel for how they perform on various courses. The session
I took was the Cayenne Off-Road Experience, because I own a
2008 base Cayenne and am interested in exploring the unpaved
portions of Texas and elsewhere. I also would like to get other
Cayenne owners interested in the same thing, perhaps putting on
off-road (or at least unpaved) tours to interesting destinations.
Anyway, the Cayenne Off-Road session consists of about an
hour on the paved courses (1-mile road course, slick track, kick
plate, and wet skidpad) and the last half hour on a custom-made
off-road section, which featured a 50-degree concrete staircase
to climb, as well as a 70-degree hard-packed dirt slope to climb
and then descend down the other side. The Cayenne (the one I
got to drive was a 2019 Turbo with the 541-hp turbo 4.8-liter
V8 – all I could say was “wow”) handled the staircase climb with
no problem, but when we got to the 70-degree slope, I hesitated,
because it looked like a vertical wall from where I sat, plus it
didn’t have off-road tires, or even all-season tires – it had lowprofile
performance summer tires on 21-inch wheels. However,
my coach told me to take it slowly and drive straight up to the
top, so I trusted him and did just that. The Cayenne handled it
like a drive to the grocery store.
However, at the top, there was an equally steep slope back
down, which was there to show how
the Cayenne’s descent control feature
worked. You put it in descent mode,
then tell it at what speed you want to
descend (in 1-mph increments), give
it a little gas, and it takes care of the
rest. My coach had set it for 3 mph,
10% Grade - 5.7 degrees
and once we started down the hill, the
Cayenne maintained that speed all
the way down to where it leveled out.
I didn’t touch the brake or accelerator
20% Grade - 11.3 degrees
pedals; all I had to do was steer. An
interesting feature for sure.
Anyway, this got me wondering
how this 70-degree slope would
appear as a “XX% Grade” sign on
mountain roads. As I’m sure you
know, the percent grade is calculated
as the vertical rise (or drop) of the
slope divided by the horizontal
Visual Representation of % Grade vs. Angle
100% Grade - 45 degrees 275% Grade - 70 degrees
distance covered, expressed as a percentage. For example, if a
slope is such that my altitude increases (or decreases) by 1 foot
for every 20 feet of horizontal distance that I travel, that comes
to 1/20 = 0.05, which is 5%. If I rise (or drop) 1 foot for every 10
feet of horizontal travel, that’s a 10% grade. If you take that to
an extreme, ascending (or descending) 1 foot for every 1 foot of
horizontal travel, that’s a 100% grade.
However, you have probably realized that 1 foot up and 1 foot
forward results in only a 45 degree angle, and that’s the clue as to
how percent grade and degree of slope are related: trigonometry!
If you took any trigonometry classes, you may recall the terms
sine, cosine, and tangent. If your memory is really good, you might
even remember the memory aid SOHCAHTOA, which keeps track
of which sides of a right triangle apply to calculating the sine, cosine,
and tangent of an angle. The TOA part means “tangent = opposite
/ adjacent,” which directly applies to our sloping mountain road.
For a roadway, the opposite side is the vertical distance, and the
adjacent side is the horizontal distance. So since the tangent of the
angle is the percent of the slope, the inverse tangent (or arctan) of
the percent of the slope is the angle of the slope.
Using our handy-dandy scientific calculator, which is an
accessory on every Windows PC, entering 45 degrees and clicking
the “tan” button yields a result of 1, which is 100%. By the same
token, entering a 1, then clicking “Inv” and then “tan” (i.e.
inverse tangent or arctan) produces a 45. Just remember that to
calculate smaller percent grades, be sure to divide the percent
number by 100 to express it as a decimal amount.
Going back to our previous examples, that 5% grade is a pitiful
2.86-degree angle. Similarly, a 10% grade is a 5.71-degree angle.
Doesn’t sound like much, and any Porsche can climb those slopes
without even breathing hard, but an 80,000-lb semi really struggles
with even just that 2.86-degree slope, as anyone who has been
stuck behind a loaded 18-wheeler on
mountain road can attest.
Which brings us back to those
slopes at the Porsche Experience
Center. That 55-degree staircase
turns out to be a 143% grade, and
the 70-degree slope would be signed
as a 275% grade if you were ever to
encounter one on the highway.
So in theory, what percent grade
is a 90-degree angle? Well, our
scientific calculator tells us that an
89-degree slope is a 5,729% grade,
and an 89.9-degree slope is a
57,729% grade, so what does that
mean for our vertical wall? The
calculator says that it’s undefined,
in other words, infinity. Sorry, but
not even a Cayenne Turbo can
handle that one . . .