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This youtube video video from How Round is Your Circle? claims you can estimate area with a tool created from a bent coat-hanger.

I am a little skeptical about the claim which is why I am posting this here as opposed to math.se.

Is this really a way to measure area, and if so what is it called (and maybe why it works)?

Oddthinking
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picakhu
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    It is now [crossposted to Math.SE.](http://math.stackexchange.com/q/45639/242) – Bill Dubuque Jun 16 '11 at 00:41
  • Planimeters are among those instruments that have to be seen to be disbelieved. They come in several varieties and can be staggeringly beautiful. A dear friend gave me a good one for my thirtieth birthday and it has pride of place among my office display pieces---yes, even ahead of my slide rule and artillery spotter's calculator. – dmckee --- ex-moderator kitten Jun 16 '11 at 01:21
  • @picakhu -- hi picakhu. Pika, pika! :) Based on your comments below you seem to **"not believe"** that a planimiter is a completely normal everyday real instrument. You can easily buy one say here http://www.engineersupply.com/tamaya-planix-5-polar-planimeter-3651-51.aspx or here http://www.amazon.com/Lasico-MECH-POLAR-PLANIMETER-ENG-CAL/dp/B001E0FHCQ. You can see many photos of it here ... http://en.wikipedia.org/wiki/Planimeter (Of course, the "cheap" planimeter in the youtube video would not be very accurate.) Yes, it is a totally normal thing. They are used every day by draftsmen. – Fattie Jun 16 '11 at 13:12
  • @Bill Tags don’t have to be specific; tags should be general enough to group subjects into categories. The tags suggested by you don’t really do this. I think even in a few years the number of questions about “measuring-devices” can be counted on one hand. – Konrad Rudolph Jun 16 '11 at 16:22
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    @konrad: Speaking as a mathematician with [decades of experience](http://en.wikipedia.org/wiki/Macsyma) in math and computation, I've encountered much skepticism about the correct function of various analog computational devices. So I wouldn't be surprised to see more than a handful of such questions here over many years time. Why throw away expert-contributed knowledge (here tags) that could later prove difficult to duplicate? – Bill Dubuque Jun 16 '11 at 16:51
  • @Bill FWIW I believe that *on math.SE*, the tags are completely appropriate. But we should take this discussion to http://meta.skeptics.stackexchange.com. – Konrad Rudolph Jun 16 '11 at 16:55
  • If I can make a simple example of why it works - if you drive a car you probably do it without thinking. If you are trying to parallel-park next to a curb, and you want to move the rear wheels to the right to be closer to the curb, you back and fill and steer the front wheels in such a way that they travel counter-clockwise around an area. The larger the area, the farther the back wheels move to the right. Basically you're integrating X over Y. – Mike Dunlavey Jun 17 '11 at 02:31

2 Answers2

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Yes, this particular device is known as a hatchet planimeter - invented by Captain Prytz. See this Encyclopædia britannica page.

For the mathematical theory of the device see Robert L. Foote: Geometry of the Prytz Planimeter. See also Mark Levi and Serge Tabachnikov On bicycle tire tracks geometry, hatchet planimeter, Menzin’s conjecture and oscillation of unicycle tracks, and Tom Apostol and M. Mnatsakanian: The method of sweeping tangents.

Bill Dubuque
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    I stumbled on a a simple explanation here .. http://whistleralley.com/planimeter/planimeter.htm (scroll down a little). – Fattie Jun 16 '11 at 13:20
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It is, the device is called a planimeter and it works because of... a whole bunch of terrifying maths

david w
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  • @david from your first source "It does not correctly measure the area when the pointer traces the perimeter of a figure", so does it work? – picakhu Jun 16 '11 at 00:07
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    Did you catch the next paragraph? "Provided that certain rules are observed when it is used, the other factors will be minimised or eliminated and the planimeter will give an adequate estimate of the area" – david w Jun 16 '11 at 00:31
  • I did, but I am still skeptical. Sorry... – picakhu Jun 16 '11 at 01:13
  • What determines the final angle? Is it just that the hatchet end drags on the paper and kind of stays put while tracing the perimeter? I see a *ton* of human error, since the end calculation is based entirely on the final angular displacement from the starting position. What makes that displacement reliable? – Hendy Jun 16 '11 at 02:36
  • Of course, I gathered the question was about wether this method was logically sound - rather than if a particular user will be able to get a reliable result from it. I think, absent human error, the 'hatchet' end will follow a [pursuit curve](http://mathworld.wolfram.com/PursuitCurve.html) and that's the basis of the method. – david w Jun 16 '11 at 03:02
  • @picakhu Hi Pica - LOL it's funny you "don't believe" in planimeters! You can see them for sale anywhere. It would be like not believing in "a ruler"! :-) The "math" is **not that complicated** in principle - you're just measuring across and down, and it estimates the small parts - in a "smaller and smaller" way. I instantly found this page with a **good explanation for non-mathematicians** ... http://whistleralley.com/planimeter/planimeter.htm – Fattie Jun 16 '11 at 13:18
  • @Hendy - Hi Hendy, your objection is a non-objection. Sure, if you begin at a really poorly-chosen point (basically a corner), it won't work. So what? You just do it a few times starting in different places, and you soon see the real area. The whole world up until about 1980 (?) was engineered and built using planimeters! Heh! – Fattie Jun 16 '11 at 13:19
  • @Joe Blow: Sigh... the links above do not specify *what* determines the angle as it follows the path -- namely, where do I point the thing as I follow the curve? It has nothing, whatsoever, to do about a poorly chosen starting point. You missed the mark. – Hendy Jun 16 '11 at 14:09
  • @Hendy, I think that is the reason I do not understand this. I mean to ask why the angle is changing, there is no particular reason to. In addition, @Joe, misses my question by a mile, I do not understand how it works, comparing it with a ruler is rather foolhardy. – picakhu Jun 16 '11 at 14:14
  • @david w: @Joe Blow's link [HERE](http://whistleralley.com/planimeter/planimeter.htm) contained the information I think I was missing. In the example shown of tracing a square, it appears that one keeps the hatchet end *on a straight line*, parallel to some prior path. Thus, in your link, I think that when one estimates the CoG and draws a line to the perimeter, the hatchet should stay on a line parallel to that line. Something like [THIS](http://i.imgur.com/0gJWA.png), where A is parallel to B. Perhaps I just missed that in all the other links! – Hendy Jun 16 '11 at 14:29
  • @picakhu: the link @Joe Blow posted is actually quite good, [HERE](http://whistleralley.com/planimeter/planimeter.htm). Scroll down and take a look at the simple example with a rectangle and note that point "B" (the hatchet end of the planimeter) is always on that vertical line. That's what changes the angle -- when moving parallel to the line, the angle doesn't change; when moving toward/away with any component of motion, the angle has to change to keep the hatchet on the line. – Hendy Jun 16 '11 at 14:31
  • @Hendy, I agree and I understand that link, what I do not understand is why the *HATCHET* planimeter works. It seems like one can ajdust the angle at will, after all, the sharp tip of the coat hanger is a pivot. – picakhu Jun 16 '11 at 15:57
  • @picakhu: the link I posted just above *is* with a hatchet planimeter (he opens with the polar type, but the example with the rectangle is a single piece, shown as the hatchet (point B) riding the line. Thus, the angle *is* determined by something. On the other hand... your video does not at all show following a line, from what I can see... so the video definitely is perplexing/incorrect if one is supposed to keep the hatchet on some imaginary line. – Hendy Jun 16 '11 at 16:01
  • @Hendy So, are you suggesting that the video is indeed incorrect? In which case, I understand what I have been misunderstanding all this while. ---Actually, even if the video is incorrect, I still do not understand it. :( – picakhu Jun 16 '11 at 16:02
  • @hendy: I don't understand what you are getting at: the displacement from the start position is crucial to determining the area, so there is no prescribed line for the hatchet end to follow. (the length of the rod * the final displacement = area) – horatio Jun 16 '11 at 16:21
  • @picakhu: not sure -- someone here who's been talking like they've used one or know a lot about them should be able to confirm/deny the video or state what determines the changes in direction of the hatchet end. In the vid it looks like they just let it free-ride on the paper where it wants. – Hendy Jun 16 '11 at 16:21
  • @pikachu: the hatchet blade is meant to prevent side-to-side motion. If you are rotating the planimeter on the pivot manually, you are creating a scraping motion. If you are doing this, you are not using the instrument correctly. As an anaology, I think you would agree that one can cause their car to travel sideways, but it is not generally considered the proper use of the tool. – horatio Jun 16 '11 at 16:23
  • @Hendy Hi Hendy - regarding your most immediate reply to me above; I'm sorry, I apparently just did not (and still do not) understand you r question, sorry. You know, I'm guessing there is some massive confusion here, ie, either you (or me!) or someone is completely at cross-purposes. May I humbly suggest you simply **try using one** and you will instantly get the idea. Sorry, I could not help in print! (Also just to be clear, I have utterly no interest in the "U tube" "video" (or whatever it is), I'm just talking about an ordinary everyday planimiter!!! – Fattie Jun 16 '11 at 16:49
  • @joe, are you for real, or just trolling? I seriously can't say. – picakhu Jun 16 '11 at 17:03