# EMBARRASSED to ask this- calculating slope

1. Apr 5, 2013

### maximiliano

....I've been calculating this for years...but for some reason thought about it today, and am having a brain f#rt!!!!

I suspect I've been mis-calculating all this time. Math help please to calculate PERCENT slope :)

Le's say I am driving (one way up a hill), and the following stats are the result of my drive :

1- Distance traveled: 10,000 feet (per the odometer)
2- Elevation gain over that distance: 1,000 feet

Question- What's my average slope PERCENTAGE??

Would I simply take the distance traveled (10,000) squared PLUS the rise (1,000) squared......and get the RUN by taking the square root of the result....which comes to ~10,050. Thus, 1,000 rise divided by 10,050 run is 9.95% slope? Or.....is it late and I'm confusing myself?:uhh:

Last edited: Apr 5, 2013
2. Apr 5, 2013

### Staff: Mentor

No squaring involved at all, just the elevation gain over the distance (times 100%).

3. Apr 5, 2013

### Mentallic

I actually had to google what percentage slope is. A 100% slope is a gradient of 1, hence the OP is just asking how to find the gradient and turn that value into a percentage.

maximiliano, you want to take the square of the distance travelled and then subtract the vertical climb squared.

The pythagorean formula is

$$a^2+b^2=c^2$$

where c is the hypotenuse or distance travelled, a can be the horizontal length and b will be the vertical length.

The gradient is found by calculating b/a, and in order to find a we do

$$a^2=c^2-b^2$$

$$a=\sqrt{c^2-b^2}$$

Hence the gradient and thus the percentage slope is

$$\frac{b}{\sqrt{c^2-b^2}}$$

$$=\frac{1,000}{\sqrt{10,000^2-1,000^2}}\cdot 100\%$$

$$\approx 10.05\%$$

4. Apr 5, 2013

### Staff: Mentor

Ah, OK, the distance is not the horizontal displacement but hypotenuse, my bad.

But the differences at typical slopes are so small they can be ignored. In this particular case it is 10.00% vs 10.05%, I doubt the measurements of the distance and elevation change justify this 0.05%.

Last edited: Apr 5, 2013
5. Apr 5, 2013

### maximiliano

Thanks Mentoric and Borek. Yea, all this time I've been just taking the "odometer" distance and the elevation gain and doing basic division (rise/distance)....like you said Borek, often the rise is gradual to the point that the difference isn't really much to worry about. BUT.....my example was just a basic example. What I am really doing is mountain climbing.....and I've been doing 20-40 degree slopes. Thus, at that degree of a slope, the difference is significant.

Well, I think I have the formula now....and will feel free to post some true average slopes without looking like a nitwit.

6. Apr 5, 2013

### Staff: Mentor

That's a different thing. Somehow I was thinking in terms of bicycle racing, where 20% is rarely seen.

7. Apr 5, 2013

### BraneChild

Don't be embarrassed, we all learn different things and at different speeds! Some of my classmates might be surprised that I don't know some basic things.

Think of it this way, a vertical slope is 100%, so a 45 degree slope would be 50%, and so on.

8. Apr 5, 2013

### jbriggs444

Mentallic gave a correct answer up-thread. 100% slope is 45 degrees. It's rise over run expressed as a percentage. Where "run" is the horizontal distance, not the distance along the slope/hypotenuse.

A vertical surface would have an undefined percentage slope.

Last edited by a moderator: May 6, 2017
9. Apr 5, 2013

### maximiliano

actually, a 45° slope is 100%. for every unit of travel, you will encounter 1 unit of rise...thus, 100%

A vertical slope would be ∞ %

10. Apr 5, 2013

### Staff: Mentor

BraneChild, this would be one of the basic things that you don't know. As already mentioned by maximiliano, an angle of 45° represents a 100% slope. Also, and angle of about 26.6° represents a 50% slope.

The slope as a percentage is really nothing more than the tangent of the angle (rise/run), so tan(45°) = 1 = 100%.

11. Apr 7, 2013

### Redbelly98

Staff Emeritus
Yes, but just remember that your odometer reading is the hypotenuse, not the horizontal run.