Hill gradients

[Blue Hills]

This got me thinking that we have several ways of measuring gradient. The most common these days is rise/run expressed as a percentage. But there's also the inclination in degrees. And back in the old days signs used to have the gradient as "1 in 3" and things like that, which when I was young I found completely baffling (and still do to some extent)*.

So here's a handy table of different values.

Angle	Pct	1 in
0°​	0%​
2.5°​	4%​	22.9​
5°​	9%​	11.5​
7.5°​	13%​	7.7​
10°​	17%​	5.8​
12.5°​	22%​	4.6​
15°​	26%​	3.9​
17.5°​	30%​	3.3​
20°​	34%​	2.9​
22.5°​	38%​	2.6​
25°​	42%​	2.4​
27.5°​	46%​	2.2​
30°​	50%​	2​

What you might find funny is that a 50% slope is 30° whereas a 100% slope is 90° Surely half way to 90° is 45°? Well, no. Although the relationship between degrees and % is roughly linear for low-ish values, it's actually a sin function, so it looks like this:


View attachment 551374

I did a bit of a search on this, and of course some people think "the old ways are old and therefore must be best". Seemingly 1 in 5 is "perfectly understandable" and 20% is "meaningless". Yeah, right.
https://www.theguardian.com/notesandqueries/query/0,,-196710,00.html

I actually think for road signs what is needed is "steep hill up" (or down). That's all the road user needs to be warned of.

thanks for that.
Yes it is mind boggling for someone who isn't a mathematician.
I would have thought percentages of 90 degrees would be something most folk could get their head round.
I'm just thankful that I live in a bit of london with some steepish slopes nearby* so I can calibrate my cycling head.
Canonbie Road, Forest Hill, Cudham just beyond Downe.

 

[Twilkes]

Slightly OT but my GPS unit shows me altitude in metres and it was fairly cheap so I'm pretty sure it doesn't have a pressure sensor. Is altitude supplied by the GPS satellites, or would the unit be working it out on its own by triangulating between the satellites it can see?

It doesn't seem that far out from what Strava would eventually tell me when I get home, but it seems very slow to respond to steep changes of altitude so I don't really trust it that much.

 

[Blue Hills]

Slightly OT but my GPS unit shows me altitude in metres and it was fairly cheap so I'm pretty sure it doesn't have a pressure sensor. Is altitude supplied by the GPS satellites, or would the unit be working it out on its own by triangulating between the satellites it can see?

It doesn't seem that far out from what Strava would eventually tell me when I get home, but it seems very slow to respond to steep changes of altitude so I don't really trust it that much.

Isn't it coming from the maps on the device?

 

[matticus]

...
I did a bit of a search on this, and of course some people think "the old ways are old and therefore must be best". Seemingly 1 in 5 is "perfectly understandable" and 20% is "meaningless". Yeah, right.
https://www.theguardian.com/notesandqueries/query/0,,-196710,00.html

I actually think for road signs what is needed is "steep hill up" (or down). That's all the road user needs to be warned of.

We should be grateful the original system wasn't based on 1 to 240.

 

[Twilkes]

Isn't it coming from the maps on the device?

No there's no mapping, just a basic bike computer with GPS tracker.

 

[Twilkes]

Yes, altitude is provided by the GPS satellites. Or more correctly, the GPS receiver unit calculates altitude, latitude and longitude by using signals from the satellites. But, for reasons that other CC'ers can explain better than I can, GPS is much less accurate at determining altitude than it is for determining latitude and longitude. That's why a barometric altimeter is useful as it's better than GPS. But it does need to be calibrated due to fluctuating pressures.

Edit: Here's a link http://www.gpsinformation.net/main/altitude.htm Old but still valid.

Interesting, so altitude should still be within about 20ish metres accuracy 95% of the time, as a conservative estimate - maybe it's down to cyclists' complicated relationship with hills that if it shows us 15 metres to the left of where we are then we just think "oh well, at least the trip distance will be about right" but if it shows us 15 metres below the height of the big hill we just climbed we're outraged.

 

[Lovacott]

Angle	Pct	1 in
30°​	50%​	2​

What you might find funny is that a 50% slope is 30° whereas a 100% slope is 90° Surely half way to 90° is 45°? Well, no. Although the relationship between degrees and % is roughly linear for low-ish values, it's actually a sin function, so it looks like this:

I did a bit of a search on this, and of course some people think "the old ways are old and therefore must be best". Seemingly 1 in 5 is "perfectly understandable" and 20% is "meaningless". Yeah, right.

I actually think for road signs what is needed is "steep hill up" (or down). That's all the road user needs to be warned of.

I've always understood the 1 in X equation to mean the number of feet up over the number of feet forward on the horizontal (although, that is only an assumption on my part).

So to me, a 1 in 2 slope would be a triangle with a 2 foot base and a 1 foot rise giving a pitch of 45 degrees.

What you seem to be saying is that a 1 in 2 is a triangle with a 2 foot hypotenuse and a 1 foot rise giving a 1.732 foot base (equating to a 1 foot rise over every 2 feet of ground travelled over).

Is that the case?

 

[Sterlo]

Some of the ones in the East Yorkshire Alps peak at almost 2%

 

[Sterlo]

This is about as bad as it gets around here (and it still knackers me!).

 

[Ming the Merciless]

This got me thinking that we have several ways of measuring gradient. The most common these days is rise/run expressed as a percentage. But there's also the inclination in degrees. And back in the old days signs used to have the gradient as "1 in 3" and things like that, which when I was young I found completely baffling (and still do to some extent)*.

So here's a handy table of different values.

Angle	Pct	1 in
0°​	0%​
2.5°​	4%​	22.9​
5°​	9%​	11.5​
7.5°​	13%​	7.7​
10°​	17%​	5.8​
12.5°​	22%​	4.6​
15°​	26%​	3.9​
17.5°​	30%​	3.3​
20°​	34%​	2.9​
22.5°​	38%​	2.6​
25°​	42%​	2.4​
27.5°​	46%​	2.2​
30°​	50%​	2​

What you might find funny is that a 50% slope is 30° whereas a 100% slope is 90° Surely half way to 90° is 45°? Well, no. Although the relationship between degrees and % is roughly linear for low-ish values, it's actually a sin function, so it looks like this:

View attachment 551374

I did a bit of a search on this, and of course some people think "the old ways are old and therefore must be best". Seemingly 1 in 5 is "perfectly understandable" and 20% is "meaningless". Yeah, right.
https://www.theguardian.com/notesandqueries/query/0,,-196710,00.html

I actually think for road signs what is needed is "steep hill up" (or down). That's all the road user needs to be warned of.

Understanding 1 in 3 etc is simply about whether you understand fractions, nothing more.

 

[Milkfloat]

From: https://assets.publishing.service.g...ile/772037/traffic-signs-manual-chapter-4.pdf

For those that need a refresher on tan versus sin

 

[RichardB]

I've always understood the 1 in X equation to mean the number of feet up over the number of feet forward on the horizontal (although, that is only an assumption on my part).

So to me, a 1 in 2 slope would be a triangle with a 2 foot base and a 1 foot rise giving a pitch of 45 degrees.

What you seem to be saying is that a 1 in 2 is a triangle with a 2 foot hypotenuse and a 1 foot rise giving a 1.732 foot base (equating to a 1 foot rise over every 2 feet of ground travelled over).

Is that the case?

Your first sentence is correct. I think of it as short for "rising one foot in every ten" or whatever. However, your next is wrong. A 1 in 1 slope is 45 degrees - one upward for every one forward.

 

[Ming the Merciless]

If you are right, and it's horizontal distance rather than actual distance travelled it means that 1 in X is even more daft and unintuitive than I previously thought.

Not really, percentages work the same way. So if it’s 20% , the height gain is 20% of the horizontal distance travelled. In other words 20 in 100 which using a common divisor of 20 reduces to 1 in 5. Basically maths learnt at primary school.

 

[Ming the Merciless]

All you really need be concerned with, is what gearing will I need for the gradient shown. This translates to which ring on the front (which I’d try and avoiding changing once in the hill) and range in back do you need. There’s no set answer as to what gearing you’ll want it’ll depend on your fitness, and fatigue at that point in time. But after tackling a few you’ll know from experience what gears to go for.

 

[DaveReading]

Not really, percentages work the same way. So if it’s 20% , the height gain is 20% of the horizontal distance travelled. In other words 20 in 100 which using a common divisor of 20 reduces to 1 in 5. Basically maths learnt at primary school.

True. But speaking as a maths tutor (of adults), I can vouch for the fact that many people have a problem converting between fractions and percentages.