anyone wondered how to convert hill steepness between degrees and %?

almikel

Active Member
Region
Australia
I used my iPhone to measure the steepness of the hill I live on, and it gave me the angle in degrees, not the %...but when hill steepness is discussed on forums etc, it's usually in %.

It's not hard to convert between degrees and %...but you'll need a smart phone/scientific calculator that can manage basic trigonometry (sin/cos/tan).

hill steepness.png

Take the above - I measured my street at 8 degrees on my iPhone - which is the X in the diagram above.
To calculate the % slope, which is rise/run x 100, if you have the degree of slope, then set the run at 1m and the calculation is:
Tan (X degrees) = Y/1m
times by 100 and you have % slope

My street is 8 degrees:
Tan 8 degrees = 0.14
100 x 0.14 = 14%

If you happened to have the % and wanted to calculate degrees, you go the the other way...
...let's say you had a 20% slope - quite steep for pavement:
20% = 20/100 = 0.2
so Tan (X degrees) = 0.2
X degrees = Arc Tan (0.2) = Tan^-1 (0.2) = 11.3 degrees

I must admit I loved trigonometry (sin/cos/tan) at school - finding "real life" uses is awesome.

Cheers,
Mike
 

scout592

Active Member
I know this is not exactly what this post is about but it gave me an idea of what to expect.
here’s a rough guide to how various gradients might feel:
  • 0%: A flat road
  • 1-3%: Slightly uphill but not particularly challenging. A bit like riding into the wind.
  • 4-6%: A manageable gradient that can cause fatigue over long periods.
  • 7-9%: Starting to become uncomfortable for seasoned riders, and very challenging for new climbers.
  • 10%-15%: A painful gradient, especially if maintained for any length of time
  • 16%+: Very challenging for riders of all abilities. Maintaining this sort of incline for any length of time is very painful.
 

christob

Well-Known Member
Ah, google! Found this:
It's actually not too complicated. Road incline grades are given in percentage, which is the rise over run percentage.
This is the same thing as the tangent, expressed as a percentage. So 100% grade means tan x = 1, where x is the angle in degrees. Thus, convert from percentage to decimal by dividing by 100, then take the arctan (also called "inverse tangent") of that to get the degrees.
Be sure your calculator is set for "degrees" trigonometric functions rather than radians or gradians.
Example: 10% grade: arctan (0.10) = 5.7 degrees.
 

Nutella

Active Member
There are plenty of free clinometer apps that will give you readouts in either % or degrees. FWIW, phone apps aren't very accurate since how you're holding your phone isn't.
 

almikel

Active Member
Region
Australia
There must be an app for that.
Could Siri or Alexa do that?
Or maybe a Google search?
I asked Siri just now and it pointed me to other websites without an answer...maybe Alexa would be better...
...and of course there's apps available that convert degrees to %...or Google searches...

...the purpose of this thread was to help people understand how to convert between degrees and %, not to just "ask Siri" for answers.

There are plenty of free clinometer apps that will give you readouts in either % or degrees. FWIW, phone apps aren't very accurate since how you're holding your phone isn't.
Agreed - but the app I used on my iPhone only showed degrees, and I was keen to know what the % was...and some basic trigonometry (using Tan degrees) gave me the the %.
I wasn't "holding" my iPhone - I placed the phone on the road at multiple points noting the angle on the inclinometer app...but I completely accept the angle read isn't that accurate.

Ah, google! Found this:
It's actually not too complicated. Road incline grades are given in percentage, which is the rise over run percentage.
This is the same thing as the tangent, expressed as a percentage. So 100% grade means tan x = 1, where x is the angle in degrees. Thus, convert from percentage to decimal by dividing by 100, then take the arctan (also called "inverse tangent") of that to get the degrees.
Be sure your calculator is set for "degrees" trigonometric functions rather than radians or gradians.
Example: 10% grade: arctan (0.10) = 5.7 degrees.
@christob - agreed - it's not too complicated converting degrees to %....that's what my original post shows/demonstrates

There must be an app for that.
Could Siri or Alexa do that?
Or maybe a Google search?

Sure there is - I was just showing how Siri/Alexa/Google would work this stuff out...and trying to convey that the math can be useful in real life...

...let's take a different example...

...I installed a besser block retaining wall at 100mm above ground level to the neighbours yard on a sloping yard.
On my side the wall was 1m at max height reducing to zero - and the bricklayer cut the top of the wall so the slope was constant rather than stepped.

I installed a timber paling fence on top of the retaining wall and wanted to know what angle to cut the bottom of each paling on my mitre saw in degrees to match the retaining wall.
With a water level it was easy to determine the height difference from the start of the retaining wall to the end (rise), and a tape measure determined the run.

As per above:
Tan (degrees) = opposite (rise) / adjacent (run)
Arc Tan (rise/run) = Tan^-1 (rise/run) = degrees to set the mitre saw at so that the bottom of each paling is cut at the same angle as the retaining wall...

...sure you could do it by "eye"...most do...but it only takes 2 measurements (rise and run) to determine the angle to set your mitre saw using Arc Tan.

Trigonometry is useful in real life...

Be sure your calculator is set for "degrees" trigonometric functions rather than radians or gradians.
Agreed - typically stay with degrees, unless you know how to use radians.
Radians are based on Pi...ie the ratio between the circumference of a circle and its diameter (3.1416....)
Pi radians = 180 degrees, 2Pi radians =360 degrees etc
Radians are used extensively in engineering and other sciences - I studied engineering at uni, and we always used radians, never degrees.

My understanding of gradians (perhaps apocryphal) is that it was an attempt to metricise a circle from 360 degrees to 400 gradians during the French revolution...
...also my understanding of where/how the 360 degrees in a circle concept came from (also perhaps apocryphal) is the (incorrectly) calculated days it took for the earth to circle the sun (actually closer to 365.25 days for 1 revolution)...
...who knows, maybe gradians would have been more successful if it was 1/1000 of a circle rather then 1/400...given 360 degrees wasn't correct either....

...regardless, I've never used gradians in either my scholastic or professional careers...I've no idea why calculators have the capability for gradians other than it's easy to have it...

...Going off topic...(hey it's my thread)...I spent a decade in the Australian Army Reserve, and we used Mils for navigation, which had 6400 graduations in a circle...a much higher resolution than 360 degrees...
...it was only after googling Mils tonight that I learned Mils comes from radians - Mils is actually short for mili-radians, and as above there's 2 x Pi radians in a circle so 2 x 3.1416 x 1000 Mils in a circle = 6283 Mils which is rounded up to 6400.
Every military compass I ever used had 6400 Mils on the dial - I had no idea before tonight there was a relationship between Mils and radians...but I like that there is.

There must be an app for that.
Could Siri or Alexa do that?
Or maybe a Google search?

Sure Siri/Alexa/Google can do lot's of things - but mostly it's good to know how it was done.

cheers,
Mike
 
Last edited:

kmccune

Active Member
Most people have little sense of things pertaining to gradient and gravity( Kind of like a foot of snow or inch of rain. Having worked in the construction industry all my life I know what percent of grade is( fall in a hundred feet) degrees of slope or fall( 45 degress equal 100% grade or ,one to one slope, roof pitch equivalents 9/12 equals 38 degrees, etc, another useful thing to know is"angle of repose" dirt generally falls off 1.5 to 1, yada, yada.I do not know of many machines that can climb a "1 to 1" slope unless its a crawler,its actually pretty easy to estimate the angle of a slope with a level and protractor.
One thing the RR people know and I know is an old standard railroad grade makes for a nice bike trail( one for a "Shay locomotive would be a bit different)
 

Alaskan

Well-Known Member
There are plenty of free clinometer apps that will give you readouts in either % or degrees. FWIW, phone apps aren't very accurate since how you're holding your phone isn't.
On Bosch powered bike that use a Nyon display, one can select Slope as a data field to show road grade in degrees in almost real time, about a 40 foot lag in calculation. - middle left field showing zero under , right under the Assist mode set at TOUR

20190901_070116.jpg