Computing Grade

One thing that has always confounded me is percent grade.
When I’m on the treadmill, I always run at 1% grade which supposedly simulates a flat surface due to the fact that the treadmill’s moving surface changes the dynamics of running slightly (versus you moving across the ground). When I train hills, I hit the up button on the grade and increase the % percent grade. Seems simple, but when I go outside to run and try to equate a treadmill grade to whatever hill I’m running on, I get confused.
It’s the same when I bike. For instance, on the Tour De France, they talk about Category 1 through 4 climbs:
In general terms, Category 4 climbs are short and easy. Category 3 climbs last approximately 5 kilometers (3.1 miles), have an average grade of 5 percent, and ascend 150 meters (500 feet). Category 2 climbs are the same length or longer at an 8 percent grade and ascend 500 meters (1,600 feet). Category 1 climbs last 20 kilometers (12.4 miles) with an average 6 percent grade and ascend 1,500 meters. Beyond category climbs include an altitude difference of at least 1,000 meters (3,280 feet) from start to finish and have an average grade of at least 7 percent.
Since I’m nowhere near a Tour De France class rider, I’m assuming that these climbs are pretty brutal, especially the Category 2 and 1 climbs. I might be able to get up them, but I certainly wouldn’t be winning any races anytime soon.
But what do these percentage grades actually feel like? Last year, I did laps on Old La Honda and it’s about 3.35 miles long and climbs 1280 ft. According to Stanford Cycling, it averages 7.3% grade.
OK. Old La Honda is tough for me and now I sort of know what 7% grade feels like. Still there are portions which feel even steeper than that.
Now onwards to the computation. I finally figured out that grade is rise in height over a given distance, and then you multiply by 100 to get percent.
So 7% grade is a rise of 7 ft. for every 100 ft. travelled. Hmmm still tough to visualize. Let’s convert that to an angle from the horizontal. So taking the arctan of the triangle formed by 7 ft. high over 100 ft of distance, that’s an arctan of 7/100, that’s about a 4 degree slope from the horizontal. Doesn’t seem like much but definitely hell to pedal!