# Who measures angles in "grads"?
**URL:** https://boards.straightdope.com/t/who-measures-angles-in-grads/290217
**Category:** Factual Questions
**Created:** [February 15, 2005, 5:05pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217 "2005-02-15T17:05:15Z")
**Posts on this page:** 20
**Page:** 2
### Author:  [@Angua](https://boards.straightdope.com/u/Angua)
#### Post date: [February 16, 2005, 11:15am UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/21 "2005-02-16T11:15:24Z")
> [@Bytegeist](#):
>
> Also, aren’t galaxy surface brightnesses usually given in _magnitudes per square arcsecond_, or something similar?
Yep – magnitudes per arcsec squared in optical astronomy. In X-ray astronomy, we use counts/square arcsec, and in radio astronomy we use Janskys/beam. Its great ‘fun’ trying to convert between the two.
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### Author:  [@BrotherCadfael](https://boards.straightdope.com/u/BrotherCadfael)
#### Post date: [February 16, 2005, 3:10pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/22 "2005-02-16T15:10:38Z")
> [@Absolute](#):
>
> And really, the choice of 100 grads as a right angle isn’t that much better than the choice of 90 degrees. They’re both arbitrary.
90 degrees is by no means arbitrary. Dividing a circle into 360 degrees allows the maximum possible divisions into whole numbers: 180 degrees (1/2), 120 degrees (1/3), 90 degrees (1/4), 72 degrees (1/5), 60 degrees (1/6), 45 degrees (1/8), 30 degrees (1/12), 10 degrees (1/36), 6 degrees (1/60), etc.
Base 10 kinda sucks, but I guess were stuck with it, seeing as we have 10 fingers.
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### Author:  [@bump](https://boards.straightdope.com/u/bump)
#### Post date: [February 16, 2005, 4:08pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/23 "2005-02-16T16:08:18Z")
> [@Chronos](#):
>
> Oops, didn’t see this before I posted.Wouldn’t that mean that neither uses milliradians? A milliradian would be (approximately) 1/6283 of a circle.
I wondered about that myself. I did some digging, and apparently, mils started out as true milli-radians, but then got simplified to 1/6400 for quick ease of use.
Here’s where I read this: [www.gwydir.demon.co.uk/jo/units/sea.htm](http://www.gwydir.demon.co.uk/jo/units/sea.htm)
It has some other (unverified) interesting mil properties, and an explanation for grads that’s interesting also.
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### Author:  [@Thin\_Ice](https://boards.straightdope.com/u/Thin_Ice)
#### Post date: [February 16, 2005, 4:35pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/24 "2005-02-16T16:35:44Z")
I was the instrument man of a three-man survey crew for about four years back in the seventies and I always measured angles in degrees, minutes and seconds. We measured distances in feet, tenths, and hundredths of feet.
When I first started, our party chief figured some math on a slide rule until we got a scientific calculator.
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### Author:  [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)
#### Post date: [February 16, 2005, 4:42pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/25 "2005-02-16T16:42:26Z")
> [@ElvisL1ves](#):
>
> If you’ve ever used, or seen, a scientific calculator, you’ve seen a key to switch angle units between degrees, radians, and grads. Degrees and radians are both pretty common, of course, but why does anyone want to divide a right angle into 100 grads instead of 90 degrees? Who uses that feature, and why?
In my work and education as an engineer, physicist, mathamagician, and marine navigation, I’ve seen decimal degrees, DMS, grads, and radians all used.
Radians are great for “pure” math, as they aren’t a “unit”, per se, requiring no conversion. (If you’ve used the Euler form of the definitions of the trigonometric functions you know what I mean.)
Degrees-minutes-seconds have been the traditional way of stating angles in engineering practice. With the advent of CAD modeling and the need to express all values in decimal terms (for the software) rather than fractions or other odd countings, decimal degrees have come into vogue, though I often still pull up drawings that use DMS.
Celestial navigation is still done in DMS, because the major calculations and tables involve “fractions of arc”-seconds. This is traditional and will probably never change, although celestial navigation is becoming less and less important. (I believe the Naval Academy has stopped even requiring it, which seems pretty dumb-assed to me. When your GPS satellites are knocked out by an EMP pulse or ASAT laser how in the hell are you going to know where you are at? There’s a reason that a sextant is part of the standard equipment on naval/commerical vessels.)
Grads have been and AFAIK are still used commonly in surveying to simplify the calculations, as already mentioned. I think for many projects, like road/bridgebuilding or architecture, the units are converted to DMS before being sent onward.
If you want to see a really effed-up set of units, look at Japanese Maritime Units.
Personally, I think we’d all be better off if we’d convert all units and counting systems to hexidecmal, but then we’d have to biologically engineer people to have three extra fingers per hand, and I don’t think either the technology or fashion is quite there year. 😃
Stranger
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### Author:  [@scotandrsn](https://boards.straightdope.com/u/scotandrsn)
#### Post date: [February 16, 2005, 5:02pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/26 "2005-02-16T17:02:18Z")
> [@Thin Ice](#):
>
> I was the instrument man of a three-man survey crew for about four years back in the seventies and I always measured angles in degrees, minutes and seconds. We measured distances in feet, tenths, and hundredths of feet.
> When I first started, our party chief figured some math on a slide rule until we got a scientific calculator.
I was going to say, I got the original survey maps for my neighborhood from the county office a couple of years ago, and all the angle measurements are given in degrees, minutes and seconds relative to the orientation of the nearby monument.
I had a calculator as a kid that enabled you to convert to grads too, but it never made sense to me why anyone would (its manual said grads were used by scientists and engineers). Was it really just an attempt, during the period of metric frenzy in this country, to decimalize EVERYTHING?
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### Author:  [@Angua](https://boards.straightdope.com/u/Angua)
#### Post date: [February 16, 2005, 5:18pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/27 "2005-02-16T17:18:06Z")
> [@scotandrsn](#):
>
> Was it really just an attempt, during the period of metric frenzy in this country, to decimalize EVERYTHING?
The gradian, or ‘gon’, or ‘grade’, was in fact part of the metric system – invented in France, and popularised during the French Revolution. [Wikipedia link](http://en.wikipedia.org/wiki/Gradian)
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### Author:  [@scotandrsn](https://boards.straightdope.com/u/scotandrsn)
#### Post date: [February 16, 2005, 5:18pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/28 "2005-02-16T17:18:51Z")
OK some brief Googling turned up a variety of stories as to where these foul gradians came from, but the most likely one seems to me to be that they were not a product of the metric frenzy of the American 1970s, but the metric frenzy of ze French 1790s. They did indeed try to decimlaize everything, including time, but not all of their units caught on. The gradian was among the casualties.
So I assume that the 1970s American frenzy for metric resurrected the grad from its complete and deserved oblivion, cementing it in the minds of GenX-ers as a very short but perplexing chapter of their past.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [February 16, 2005, 6:21pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/29 "2005-02-16T18:21:46Z")
> [@](#):
>
> It has some other (unverified) interesting mil properties, and an explanation for grads that’s interesting also.
If you’re referring to the relationship between angular size, actual size, and distance, yes, that’s true (as long as you’re using true milliradians). An object with a true size of one meter at a distance of one kilometer will subtend an angle of one milliradian. This is because radians are defined in terms of the length of an arc of a circle, and for very small angles, the arc of a circle and the chord with the same endpoints are almost identical. On the historical points, I cannot say.
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### Author:  [@mojave66](https://boards.straightdope.com/u/mojave66)
#### Post date: [February 16, 2005, 7:20pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/30 "2005-02-16T19:20:47Z")
The most useful thing I’ve found for the gradient is that it can covert road grade into radians or degrees. The percent grade is the same as gradient.
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### Author:  [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)
#### Post date: [February 16, 2005, 8:12pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/31 "2005-02-16T20:12:39Z")
> [@mojave66](#):
>
> The most useful thing I’ve found for the gradient is that it can covert road grade into radians or degrees. **The percent grade is the same as gradient**.
Nope. The grade is the **rise** / **run**. I was going to say this might apply for small angles, but on checking that isn’t even the case for any practical application. The percent grade is the same as the tangent \* 100. A 12% grade is 7.6 grads. It’s a frequent misconception, though, and I can’t count the number of engineers who have assumed that grade is roughly equivilent to angle.
For calculations involving radians, the [small angle approximation](http://maxwell.physics.mun.ca/MPL/Physics1054/Ancillary/SmallAngles.pdf) (angle in radians = tangent) holds true to about 1% to 7\*pi/90 (14 degrees), and is useful in reducing non-linear differential equations to a first-order linear problem. (The classic example is the swinging pendulum, but it reoccurs all over in dynamics, vibrations, and particle physics.)
Stranger
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### Author:  [@Stranger\_On\_A\_Train](https://boards.straightdope.com/u/Stranger_On_A_Train)
#### Post date: [February 16, 2005, 8:34pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/32 "2005-02-16T20:34:55Z")
> [@Stranger On A Train](#):
>
> For calculations involving radians, the [small angle approximation](http://maxwell.physics.mun.ca/MPL/Physics1054/Ancillary/SmallAngles.pdf) (angle in radians = tangent) holds true to about 1% to 7\*pi/90 (14 degrees),
Clarification:
The small angle approximation is accurate to within 1% of the difference between the angle (expressed in radians) and the tangent of that angle, for angles up to 14 degrees.
If this were not the case, we’d have great difficulty in modeling a lot of mechanical behavior involving combined rotational and linear motion. Anybody who is trying to learn trigonometry should study [Euler’s Formula](http://mathworld.wolfram.com/EulerFormula.html). It is one of the most beautiful relations in mathematics, and once you understand it you’ll never be confused about what sines and cosines mean again. (It also makes solving integrals and anti-derivatives _sooo_ much easier than memorizing a bunch of relations and half-angle formulas.)
The first time I saw the identity e[sup]i\*pi[/sup]+1=0 was when I understood how amazing all these little symbols and equations could actually be.
I know, you think I’m sick. :rolleyes:
Stranger
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### Author:  [@QuizCustodet](https://boards.straightdope.com/u/QuizCustodet)
#### Post date: [February 16, 2005, 8:56pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/33 "2005-02-16T20:56:39Z")
> [@Chronos](#):
>
> If you’re referring to the relationship between angular size, actual size, and distance, yes, that’s true (as long as you’re using true milliradians). An object with a true size of one meter at a distance of one kilometer will subtend an angle of one milliradian.
I’ve spent much of the last four summers as the Command Post Officer of a UOTC (like American ROTC) artillery battery - we do indeed use mils as 1/6400 of a circle. I’m puzzled by **bump** ’s assertion that European mils are 1/6000th, as I’d be willing to lay money that the size of a mil would be a NATO standard, to save really horrendous confusion between say, an American forward observer and a French artillery battery. I’ve never seen a non-British battery, though, so I can only speak for the Royal Artillery.
Re: 6400 vs 6283, it does make calculations slightly easier and on the scales artillery fires on (a good FOO should never be correcting by more than 800 m, at a maximum distance of about 5 km from his position) the 2% error involved in the approximation is well within the kill radius of a high explosive shell.
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### Author:  [@mojave66](https://boards.straightdope.com/u/mojave66)
#### Post date: [February 16, 2005, 9:53pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/34 "2005-02-16T21:53:23Z")
> [@Stranger On A Train](#):
>
> Nope. The grade is the **rise** / **run**. I was going to say this might apply for small angles, but on checking that isn’t even the case for any practical application. The percent grade is the same as the tangent \* 100. A 12% grade is 7.6 grads. It’s a frequent misconception, though, and I can’t count the number of engineers who have assumed that grade is roughly equivilent to angle.
>
> Stranger
D’oh. OK, I made a blunder by from a formula that used road grade in their calculation (as Theta), and stated that “Use the degree of the road angle in Theta. Close enough.” So of course my brain says “well, 90 is close to 100, ergo…”
Never Assume. It makes an Ass out of U and ME. \
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### Author:  [@Timz](https://boards.straightdope.com/u/Timz)
#### Post date: [July 27, 2025, 7:53pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/35 "2025-07-27T19:53:33Z")
Years ago, the latitude-longitude grid on French topo maps was in grads. Maybe still?
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### Author:  [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)
#### Post date: [July 27, 2025, 11:39pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/36 "2025-07-27T23:39:58Z")
In school I was told it was for when measuring grades because 45 degrees is a 100% grade. Which would make sense if there were 800 grads in a circle and not 400.
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### Author:  [@Jaguars](https://boards.straightdope.com/u/Jaguars)
#### Post date: [July 28, 2025, 3:11am UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/37 "2025-07-28T03:11:17Z")
I can vouch that surveying in NZ, Australia, Canada and the US all still use Degree/Minute/Second. I believe the UK follows suit.
The only time I’ve run across Gradians was at tech, where one lucky group was supplied with a old Swiss-built theodolite and were stuck doing their project in a totally different angular system.
I’m not aware of any relation between Gradians and engineering gradients, but then my education was very much on the applied end of the spectrum.
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### Author:  [@DPRK](https://boards.straightdope.com/u/DPRK)
#### Post date: [July 28, 2025, 4:11am UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/38 "2025-07-28T04:11:15Z")
The gradi **ent** of a function is like its slope (direction and rate of fastest increase). So the value is more like the tangent of an angle, not the measure of an angle.
If you look at a modern nautical almanac (not a super old one) or other navigational material, you will note that the entries are given in degrees and decimal minutes, e.g., 231° 56’.1, because no way are you measuring arcseconds using your sextant.
Astronomical data will have degrees, minutes, and seconds, though.
> [@Saint\_Cad](#):
>
> In school I was told it was for when measuring grades because 45 degrees is a 100% grade. Which would make sense if there were 800 grads in a circle and not 400.
Doesn’t matter how many grads are in a circle. 90 degrees is not a 200% grade…
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [July 28, 2025, 12:08pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/39 "2025-07-28T12:08:52Z")
There is a linear relationship, but it’s only approximate, and only works for small angles. And the units it works with are radians, not degrees or grads. If you have a 5% grade, that really is an angle of 0.05 radians.
But yeah, by the time you get to a 100% grade, that relationship is out the window no matter what units you’re using.
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### Author:  [@GargoyleWB](https://boards.straightdope.com/u/GargoyleWB)
#### Post date: [July 28, 2025, 3:23pm UTC](https://boards.straightdope.com/t/who-measures-angles-in-grads/290217/40 "2025-07-28T15:23:28Z")
I’m an engineer. Most/all of the CAD design software I’ve used default to decimal degrees. I.e. 23.75 degrees instead of 23 deg 45 minutes.
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