A tool to measure the area of any shape

A polar planimeter, which measures the area of a region when you trace around it.
This is episode 9 of my video series about calculating devices.
My video about doing calculus with a planimeter: • Integrating with a pla...
End song inspired by "Hotter than a molotov" by The Coup.
Chris Staecker webarea: faculty.fairfield.edu/cstaecker/

[@OwenHodges]

I worked construction one year, the accuracy of my coworkers astounded me. With a lot of practice your hands can be very precise. I'd say user error is at fault.

[@garylin334]

Agreed. Haha. I have one. Thanks for showing me how to use it. I think the tracer bar is supposed to be set at 89.7.

[@PotatoClips]

Just to add to that, you can take a few readings and average them to make good use of the high precision. The more samples you take, the more the user error will mostly cancel out over time (unless you have a shape where people tend to overshoot the same way in one spot more so than in other spots).

[@anteshell]

@@PotatoClips The same error in the same place is easily mitigated by turning the measured paper on different angle or simply by just repositioning yourself to other side. The same mechanical inaccuracies would still apply but making the same error in the same place would be eliminated.

[@rakau123]

I've used this. Best way to minimize error is to scribe the circumference 3 times in one go and dividing the answer by 3

[@thefekete]

You can easily feel a 0.001 inch difference in adjacent features of machined parts.. amazing what these flesh bags are capable of.

[@TaylorTheOtter]

I saw this in an antique shop some time ago and I tried to figure out what it was. I never found it until now. I'm gonna go back to see if it's still there because this is way too awesome.

[@joen7xxx]

I have used these before computers came into engineering offices. I still have this exact model and actually used it last year.

[@ChrisStaecker]

What did you use it for last year?

[@aajpeter]

Waiting with baited breath

[@joen7xxx]

To determine the area of a mine waste rock facility for reclamation

[@aajpeter]

@@joen7xxx awesome, my hero! Necro thread success!

[@naajohnnorthcott8267]

There are ways of improving accuracy: position a straightedge so that the pointer can slide along it to trace straight portions. This requires experience. Always do the wiggly bits first, if your hand slips you have wasted less time. Measure until you have at least three close to identical results. Never zero it - this will increase wear around zero - it's not difficult to subtract two readings. Never use the second reading from one measurement as the first of the next measurement - you could be carrying over an error. Once you've mastered all this, how about inventing something to measure volume. Without using water.

[@meferswift]

We can use oil instead

[@jeffreycrawley1216]

I used one to measure the volume of the High Island dams built in Hong Kong in the late 1960s/early 70s. Cross sections were drawn at regular intervals, the areas measured and then the mean of two adjacent slices multiplied by the interval - it took SUCH a long time!

[@trollmcclure1884]

@@jeffreycrawley1216 imagine calculating planetary orbits from a moving and rotating planet 🙂. I think you must sign some papers to a devil to predict a path of a solar eclipse 20 years from now. Or to shoot a rocket at Pluto

[@jeffreycrawley1216]

@@trollmcclure1884 What on earth are you talking about? A planimeter has nothing to do with the prediction of the paths of planets. It's used to measure an area on a plane (ie flat) surface represented on a drawing or map.

[@trollmcclure1884]

@@jeffreycrawley1216 ... which is what? Less complicated than the other thing. Awesome

[@StewartRussell]

The calibration compass (the flat thing in the bottom right of the case at 0:13) rather cleverly engages with the planimeter's viewing loupe to trace out a circle of exactly 10 in². Really diligent map-tracers would trace round a feature multiple times, then divide the result to get a reading with the errors averaged out. Now we just right-click on a GIS feature to show the properties … They're neat tools. Prytz (or hatchet) planimeters are even simpler devices - a point and a blade on a beam of known length - yet give a fair approximation of area. They're even more fiddly to use than a polar planimeter, but when you gotta measure the area of a Strawberry Shortcake character's head like now, they'll do the job

[@Jeremy.Bearemy]

It seems like every day I'm spending more and more time measuring the area of Strawberry Shortcakes head. I wish I could get a tool to make it more efficient

[@ccreutzig]

These days, of course, you can simply make a photocopy of Strawberry Shortcake, cut the shape out and weigh the paper ...

[@giovannicesaramorim9adigan961]

@@ccreutzig Genius, and then divide by the grams/m²

[@RagtimeBillyPeaches]

In order to minimize the error you run the 'pointer' around the shape three times and divide the measured area by three. I have a very old Lietz planimeter, marked 'A. Lietz Co. San Francisco U.S.A.', and 'Swiss Made' (Serial No. 59876). It is not adjustable for scale, and you must rotate the scale wheel in order to Zero the device. The radius distance from the wheel to the pin is only 5.6 inches, and one must divide the shape to be measured into small enough areas to be traced without moving the pin, or over-centering the arms. I received the planimeter as a gift from a Survey Party Chief that I worked with, at Cook Associates in Oroville, CA, in 1969. Since I don't have AutoCad, I still use it these days when designing sailboats.

[@theinspector1023]

It's possibly a planimeter for steam engine indicator diagrams. I think they're usually smallish and non-adjustable. Mind you I'm probably wrong.

[@abnormaalz]

I find it absolutely insane that there were people that were knowledged about these mathematical concepts, able to create such devices and actually went through the pain of making such a thing AT THE SAME TIME, all that back in those days.

[@davemiii]

Thanks for posting this, this is the one that brought me to your channel. I'm researching my great-grandfather's Lippincott planimeter from the early 1910's. He was a steam engineer when most large factories ran on steam engines and belt drives, etc. It used glass rods that you would install with a slip-on wheel. The glass rods were hollow with a scale on the inside (1/20, 1/30, 1/40). I also have his pantograph which was used to copy the output plot from a steam gauge on a cylindrical wheel that rotated as it was pulled by a string on the output of the piston. The gauge had a stylus attached to a pressure gauge so as the piston expanded and retracted and the pressure rose and fell, it would generate the PV curve for the steam engine. The plot would be removed and he would use the planimeter to calculate the power output. A lot of detail in my comment, I know, but I thought you'd be interested. I'm still trying to figure out how the rods and scales were used. Thanks again.

[@davemiii]

Here's the application. Thought you might be curious. Gauge is at about 6:30, plots are shown early on and at the end though. Thanks again. kzbin.info/www/bejne/aYuaaKSjnbeNZ6M

[@ChrisStaecker]

Sounds awesome- it's crazy all the contraptions that people came up with before the digital age. Thanks for watching-

[@rayomarte9412]

Found this instrument in my grandfather's chest of goodies. Finally found out what it was and your video explains how to use. Thanks!

[@drawbar217]

I was taught that it is standard practice to measure a polygon four times in each direction and average the results to minimize the human error. i still have my fathers K+E in its pristine box.

[@allanrichardson9081]

Since the wheel, and thus the dial, would roll backwards when going CCW, I assume that tracing the same area in reverse would bring the reading back to where it started. So your procedure should not only give two forward measurements to average, but also show the accumulated error when finished!

[@jetison333]

@@allanrichardson9081 I believe by each direction the commenter meant to rotate the drawing four times, 90 degrees each, but tracing backwards might not be a bad idea either.

[@kabzebrowski]

Just watched a video on mechanical integrators, nice to know such tools can actually be built - I only knew about using the scale trick to measure areas (draw your unknown area in paper/wood and weigh it against known squares of the same material and thickness). I think we got too ingrained into thinking of squares/ish things, and discrete computers, so it's cool to know machines can do curves too, and that we're again going back into researching about analog computers.

[@geoffcrumblin7505]

I used one of these around 30 years ago. We used this to verify the design volumes of plastic bottles that I designed.. The drawing was composed of many cross sections, as the bottle changed shape throughout its length. The drawing was then passed to a patternmaker, who would produce a wooden 3D model that could be checked by displacement in a water tank. Now of course, its checked within the CAD system as its being created.

[@americannomadnewsthecardbo4339]

I had to stop the video at 5:19 to point out that development of a theorem and proof of the theorem are two separate functions in the society of mathematics. One may develop a theorem and then go about understanding whether they have developed a theorem which is correct or incorrect one must proceed the other.

[@stanleydodds9]

I think the more important fact is that Green's theorem is a lot more general than what is needed to prove that this device works, or to know how to make one. Pretty much all you need to know is the area sine rule for triangles, area = 1/2*a*b*sin(C), which is equivalent to other simple and well known facts about vector cross products. One side length is the distance from the pivot to the tracing point, the other side length is a small distance you move it in a given direction, and the angle is the angle between these two lines. Once you know this, understanding or designing such a device is not too difficult and you need nothing close to the generality of Green's theorem.

[@zefdin101]

What a great, great explanation of this device. Thank you Chris!

[@maximumiq158]

I got one of these in my shop. Never actually had to use it, but its amazing to calculate the area of any shape with such a relatively simple mechanical construction.

[@melody3741]

I don't even know how to describe this, but somehow your commentary is the most perfect commentary that commentary that could have ever been on this video of this device.

[@moosekababs]

this is actually so cool and i cant wait to check out other stuff from you, thank you for helping me learn so much neat stuff, it means so much to me!!!!!

[@kyler_funk]

Wish these videos were a couple times a week, the content is unmatched like nothing else out there!

[@mcrsit]

Subbed. Your sarcasm deserves more, and I deserve more of it :P

[@BoomedYetLush]

The final comment is some of the best advice I have heard in a while.

[@paulayala4816]

I used to use a planimeter a lot when I first started in engineering. It worked well for figuring out the areas for sections (even sections with different horiz and vert scales) as well as contour maps, either way could be used to calculate earthwork volumes. For additional accuracy we would often go around an area 2 or 3 times, read the area then divide it by the number of times you went around it. It is so much easier now to figure out earthwork using Civil 3D, still there are times I wish I had a planimeter to avoid the setup involved using Civil 3D.

[@puskajussi37]

Grats! This really deserved to become picked up by the algorithm. So fascinating!

[@alimacdee]

Brilliant, right to the point & funny, I wish all KZbin videos were like this!

[@atamo8770]

I had the idea and wanted to make such a device. I am happy that someone actually made it.

[@skykadet3550]

The video was great! But the end is what sold me. Subbed!

[@lotgc]

Wow this is so cool! How come no schools have these? This would have made geometry a lot more interesting

[@E12345E]

im going to recommend this to my geometry teacher, maybe we could use this sometime

[@StrangeGamer859]

I'm guessing they are really niche, hard to find tools

[@lotgc]

@@StrangeGamer859 I suppose, but the tool doesn't seem particularly complicated. I bet you could make a pretty simple one at home with some popsicle sticks of you knew what to do

[@ChrisStaecker]

@@StrangeGamer859 They are easy to get on ebay. You can also get brand new digital ones, but that's a bit less fun. There is a simpler type that you can make yourself called a "Prytz" or "hatchet" planimeter.

[@xenn4985]

Because no job uses them, its an obsolete technology.

[@gabrielh5105]

This can actually be done thanks to the Stokes theorem (and the generalized stokes theorem) in calculus, which, in very few words, relates the area of a shape with its perimeter

[@wilhelmching5370]

Awesome! Informative and entertaining at the same time :)

[@paulwyleciol3459]

Absolutely incredible, thanks for posting!

[@NVRMTmotion]

I see these come up in auctions fairly regularly. They're always so tempting even though I have no real usecase.... Yet!

[@dhy5342]

Scientific American once described a much simpler version of this tool. Simply get a length of stiff wire (they used a section of a wire coat hanger), hammer one end flat and sharpen it like a knife blade. Bend a couple of inches of this end 90 degrees so that the blade part is parallel to the length of the wire. Then sharpen the other end of the wire to a dull point and bend the end to match the other end. Hold the resulting device so that the blade is resting against the paper on which the area to be measured is drawn (mark this spot) and trace the outline of the area being measured with the pointed end. The area will be found by multiplying the distance the blade has moved from its original position times the length of the device from the blade center to the pointed end. Described here: persweb.wabash.edu/facstaff/footer/Planimeter/Prytz/Prytz.htm

[@James_Hallam]

I have one of these, purchased new in the late 90s. Pretty sure you can still buy them. They are used in yacht design to measure the areas of hull sections, which are then used to calculate important things like hull volume, displacement, waterline and immersion rate.

[@hamletksquid2702]

Thanks. I knew it had to be useful for something. Now I have an excuse to get one.

[@risvegliato]

you can indeed stil buy them - check out the tamaya planix 7 digital planimeter. I work in a calibration lab and we get these sent in for calibration - we check them using graph paper! (the graph paper is checked against an accurate scale)

[@nop3noperson]

Seriously this channel is teaching me things.

[@ianbell8701]

Great trip down memory lane. I used one of these early in my engineering career. One can only measure areas if the weighted fulcrum is outside of the area being measured…at least that is what I remember. I was a an aircraft stress engineer and if you wanted to know the shear-flow around a complex cross-section you needed to know the enclosed area of the cross-section (shear-flow = torque / twice the area, q=T/2A). The planimeter is the physical manifestation of calculus of the definite integral.

[@theprojectproject01]

Hol' up. What you just said? My brain translated that as your effectively saying, The math demands that the tool be thus-and-such a shape. Now, I know that isn't literally true, but it feels... directionally true? Do you grok what I'm rappin' about, man?

[@ianbell8701]

@@theprojectproject01 All I said was that the weighted (fixed) fulcrum must be outside the enclosed shape being measured.

[@theprojectproject01]

@@ianbell8701 Oh, I know, but it kind of broke my brain, maaaaan

[@drakeolejniczak2357]

This video is amazing. Thank you.

[@envitech02]

Wow!! This is so cool and astounding!!! It's awesome!!

[@scheimong]

My mum studied architecture in the late 80s, so she has a giant box of tools from her university days. I remember reorganising our old belongings one day when I found one of these collecting dust in the box. It wasn't as fancy as this one is the video, but young me found that thing absolutely amazing. My mum showed me how it works but didn't let me play with it any further, since I probably would have destroyed it 🤣

[@lourias]

I would sure love to see how the dials and gears interact. Such a cool tool!

[@ChrisStaecker]

No gears! (except one to turn the wheel for the higher digit) The dial that reads the answer is in direct contact with the table, not mediated by any other gears.

[@johneygd]

Incredible simple but very effective.

[@barunrahiroy5762]

Thank you for uploading this video

[@GH-oi2jf]

I learned about this when I got a summer job in a surveying office. I thought it was amazing.

[@aggrontv1450]

very well put together video, keep it up!

[@RalphReagan]

I love your channel!

[@MattMcIrvin]

I spent a long time trying to figure out if I could somehow intuitively understand how this works. The Wikipedia article on planimeters actually has a pretty good basic explanation of how a *linear* planimeter works. That's a thing that is just like this except that the outer half of the arm is attached to a sliding carriage on a straight track, on which it can pivot. They describe how it measures the area of a rectangle, and I guess you could conceptually break a shape down into rectangles. The key thing is the difference in the angle of the measuring wheel as it slides along the nearest vs. the furthest edges (because the other edges should cancel). I suppose that explanation would also work for a polar planimeter if you changed it to a kind of keystone-shaped wedge in polar coordinates, centered around the fixed pole. And then you could think of everything as built out of those.

[@scottmackenzie7207]

Super helpful! I saw one of these for sale and had to know more. BTW, if you were doing a gravelled old man's cartoon voice you'd sound EXACTLY like Rick from Rick and Morty 🤣. Keep em coming, I'm subscribed!

[@poobertop]

You just measured the perimeter of my interest, turns out it's subscribed to the power of a like.

[@larrylawson5172]

The USDA Agriculture and Stabilization and Conservation Service (ASCS), now Farm Service Agency (FSA), used to use these to measure farmer's fields for various programs. How many acres of a crop. How many acres of a waterway. Area in acres of a pond. And so on. This might be used to determine a payment for some program. The maps were all at a very precise scale so that the planimeter would measure acres. For accuracy you always measured 3 times and then divided the answer by 3. The planimeters did not have the adjustment scale so it was very important that the adjustment for scale be done in the production of the maps. Acreages were considered accurate to the 1/10th of an acre. Later they used an improved electronic version that would let you mark corners but also follow irregular shapes. Later yet came GPS. If I recall correctly, the map scale was 660' to the 1/8 foot. 660' was an important number. It was 10 surveyor's chains or 40 surveyor's rods long. A surveyor's chain is 66' long. A surveyor's rod is 16 1/2' long. A square mile is 80 chains by 80 chains or 6400 square chains or 640 acres. And that is another interesting math story. It involves the English, horses and plowing (ploughing) and the need for farmers to know area.

[@RalphReagan]

One day our chain broke so we finished the survey in chains and links. (It broke at 66 ft). The computer guy had a fit.

[@larrylawson5172]

@@RalphReagan He had long wavy hair prior to the chain break and now has less hair than a que ball.

[@davidnyoike6996]

The outro is lit🔥🔥

[@JustinClarkAGolfo]

WTD DUDE I ABSOLUTELY LOVE YOUR VIDEO. You're the funniest ever.

[@johneygd]

Me too,, except his stupid annoying smart jokes at the end.

[@ChrisStaecker]

Harsh! Bro... I’m right here...

[@JonathanVaucher]

The vernier is actually very usefull. If you want to measure areas in square centimeters (cm^2), put the measuring arm vernier at 149,5 mm. Then every number you read on the roller wheel (big graduations) is a multiple of 10 cm^2 Every small graduation on the roller wheel (smallest graduations) is a unit of 1 cm^2 The vernier wheel enables you to measure decimal unit (aka multiples of 0.1 cm^2) For the flat disk, each number represents 100 cm^2 For exemple, with the measuring arm vernier placed at 149,5 mm, your reading (see video at 2:40) would represent 342.2 cm^2 (quite hard of an area to measure with this device as you would need a longer pivot arm). So in truth if you are to measure in the 0 to 10 cm^2 area range, having a vernier sur doesn't hurt to have one more significant figure in your measurement. I use this device to measure in the range of typical 1.0 to 4.0 cm^2 area ranges on histologic slides. I need to measure area of stained lung tissu on the histologic slide as I need to make counts of the number of asbestos fibres per cm^2 in order to be able to make diagnoses of asbestos lung disease under the microscope (diagnostic criteria are diffuse parenchymal fibrosis AND at least 2 asbestos fibers per cm^2 of lung tissu on stained histologic slide). This device helps me so much !

[@photelegy]

Can you please show how to measure the area of the Mandelbrot form? Thank you 😉

[@dsamok]

the way you explained is just awesome, it was immensely helpful, thanks so much

[@HexaMelonN]

The ending lmaoooo, also really cool tool, I would love to try one

[@daikucoffee5316]

This is the coolest thing ever

[@alleballe903]

Cool thing my friend!!

[@cdorcey1735]

At 3:52, we see a mechanism at the top of the picture with a clamp at the left and an adjustment wheel, which allows precise alignment of the mechanism with the arm. There's a second vernier scale at the upper right, so this thing was built for plenty of precision. At the right, it looks like the setting is "9.18". 8" x 0.918 = 7.34. At 2.55, you measure 7.7 when you were expecting 8, but maybe you should have been expecting 7.34.

[@allanrichardson9081]

If the tracing window had a tiny hole at the intersection of the cross hairs, you could use a dried-up ball point refill poked through the hole to trace the figure more accurately.

[@josh9135]

I found this to be insightful and humorous

[@wmfwoodworking]

Fantastic. Subscribed.

[@omarhassan6522]

thanks for this super great video

[@sebbes333]

AWESOME! I feel kind of sad that I never have had (and probably never will have?) a use for this device... :/

[@EPMTUNES]

this is really very cool

[@StephanBuchin]

Great channel ! ☺

[@valentinsarmagal]

Good sense of humor bro 😂😂👌👌👌

[@chris-hu7tm]

love the music

[@gauravsimha]

Dude you're hilarious man ! Great video.

[@ChrisStaecker]

Thanks! This video's been getting more views lately, not sure why- click my channel for lots more.

[@patriciomassun]

jajajajaja I love that ending! I´m suscriptor for now!

[@LimeyGuru]

I was given one of these about 60 years ago by an old engineer who told me that it was used to measure the area of a steam engine dynamometer curve to determine its horsepower.

[@jeffreycrawley1216]

Integration of the area under a curve - I hated calculus!

[@davidbostock6933]

I recall glancing through use instructions many decades ago. Useful in one direction, more precision was involved in tracing in the other direction, it seems they would have been averaged. Vernier is easy to use and figure out why it works, just look at it carefully. Maybe look for pdf photo of original instructions.

[@MrKotBonifacy]

Learned how to use this gizmo in sec. school. Like, nearly 40 yrs ago... Ah, sweet memories... ;-)

[@BornAgainstAll]

Amazing video, when I saw what this could do I immediately started running through the mechanics of it and how that would translate into math. So I was already hoping you would give the formula, which oddly enough to say made my heart skip when you showed it. Your thoughts on mathematics completely took me by surprise afterward because of how much it resonated with me. I should have paid attention in math class during school....

[@FlatLineStudios]

@5:34 ...what you are about to say... ...yes! Absolutely.

[@websitesthatneedanem]

Awesome! 👍

[@Aloha_XERO]

As suggested: Liked : [ 👍🏾 ] & Subscribed : [ ✅ ]

[@docbones213]

That's rad.

[@InnerBushman]

I believe the error was so large because the sliding scale on this device was in some arbitrary position. Set it to exactly 10 and try again, 3 times in a row, then sum up the results and divide by 3. Let's see how much of an error you get then.

[@ericfermin8347]

I'm stealing your last line for the next time I'm teaching series in Discrete math.

[@gonderage]

that ending monologue was pretty funny tho

[@geniemaster5907]

Thanks. Made me wiser. :)

[@user-du2of3lh1g]

You should do a video about a sector it's an interesting tool!

[@ChrisStaecker]

yes! Been planning to do one for a long time. Someday...

[@YYYValentine]

This is genious!

[@davidmam]

would you not calibrate by comparing the measured area to measurement of a known are and scaling it? the adjustable beam then lets you set the scale to match the size of the feature being measured. the numbered scale and vernier adjustment lets it be set up consistently between uses

[@ChrisStaecker]

Yes there is a calibration procedure and there was a big body of conventional wisdom about the best way to get accurate measurements. Needless to say I am not an expert in these techniques!

[@jeffreycrawley1216]

My planimeter has a metal test bar that will describe an accurate 10sq inch or 19.94sq inch circle so you can check the calibration.

[@jmjuzd8449]

Informative.

[@danielpakk]

You forgot a dxdy in the surface integral. I teach calculus... Thank you so much for all the videos!

[@ChrisStaecker]

I also teach calculus... you're right!

[@payno6643]

this guy is hilarious subscribed

[@markusfischer856]

How much does it cost now?

[@LZRCuteR]

Need this, not sure why

[@peterfitzpatrick7032]

You can buy brand new Mitutoyo vernier calipers which can measure in 0.02mm increments... I regularly use my Moore & Wright vernier in the shop when machining... 😎👍☘🍺

[@kurumi394]

I guess this is a real world application of Green's theorem which says that a line integral along a perimeter and double integral are the same thing? Fascinating little device.

[@ChrisStaecker]

Yes! There's a video on youtube somewhere that explains the connection a little bit and that video is this video right here that you are commenting on.

[@igoranisimov6549]

Looks like it uses integration in polar coordinate system, angle and radial distance. That explains it.

[@TiagoTiagoT]

What's the purpose of the marks on the metal arm and on the little piece next to it on the device?

[@GH-oi2jf]

Possibly to calibrate it to a particular scale. When I used one it had no such adjustments, but I used it on aerial photographs which were made to a fixed scale.

[@giovannifontanetto9604]

Just took calculus 3, immediately saw the connection.

[@egomez3337]

Wow, niece tool.

[@leeself8361]

At 2:10 "You trace the figure going clockwise," proceeds moving the tracing arm counter-clockwise. Unless the image is inverted between left and right. But joking aside that is a nifty tool.

[@ChrisStaecker]

I’m measuring the hat.

[@BrokenLifeCycle]

These old tools and analog computers are so fascinating. Too bad they're not produced anymore. I mean, try finding a good Slide Rule in this day and age.

[@davedave6010]

On most callipers*... It's used way more than you think ... Mostly in schools or firms without the money to buy digital ones

[@Woodlawn22]

ARGHH...measuring earthwork quantities from grading plans typically with one or two foot contours. Hours and hours of my life were spent using one of these as a civil engineering design drafter.

[@Ttavoc]

Vernier-Scales are used on calipers up to today. Nearly each mechanical quality caliper has such a scale.

[@ChrisStaecker]

See my "Caliputer" video for another discussion of Verniers.

[@tombouie]

Amazing

[@loliloloso]

Thank you I, I never heard of this devise. I think it’s not Green Theorem but possibly a special case of it, the polar coordinate integration formula: A= 0.5 \int r^2 d\theta . This one should be known to Newton, Kepler’s 2nd law can be derived from it.

[@loliloloso]

The story of George Green is interesting, too. He was a working class man, a self-taught amateur who made important contribution to mathematical physics.

[@s1gm4_4c4d3my]

The formula you mentioned can be derived from another formula obtained via the Gauss Green theorem: half of the integral along the border (going anticlockwise) of the differential form x dy - y dx. Parametrizing the border with \theta you get A = 0.5*\int x(\theta)\frac{dy}{d\theta}-y(\theta)\frac{dx}{d\theta} d\theta and then knowing x(\theta)=r cos(\theta), y(\theta) = r sin(\theta) you get the formula you mentioned

[@loliloloso]

@@s1gm4_4c4d3my : thanks, you are right, the equality you said can prove Kepler's 2nd law, should be known perhaps by Newton, long before this devise was invented.