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Hi Everyone, There seems to be a lot of talk today about Area and Volume. Most of it is computer Programs. And that is great,Survey tech, is moving at the speed of light. That is fast 186,242 miles per second. All surveyors should know this. Why that is how fast your GPS works from the satellites to you, so how many seconds does it take at 12,500 miles. What is that ,that how far the satellites orbit around Earth. How many of you Knew this. I hope a lot . The same as how old is Sun light, it is 93,000,000.00 million miles away. All this math should be easy for us, Just as how old is Moon Light it 250,000.00 miles away. It is easy to just go to, Just asked .com, for all these answer. But how many square feet on a lot or tract of land you are doing. That they can not help you with. And all you survey students , you have to take a test. No laptops there. And i am doing this with out a calculator program.

But look at what i am about to show you, this can be done on a calculator . And that is how you will do it on your Test. This is Area By Coordinates. I will show you 3 different ways. Two of which is just a reversal

of the other one, but X always checks Y. So here is Area by Coordinates.

AREA BY COORDINATES with out a Program

Given a closed figure defined by points of known coordinates ( N x ,E x) the figure area can be determined by cross- multiplication of the coordinates pairs. I am using  US  Feet, all of this will work with meters.
As the same, If you are using feet you values will be in feet values, If you are using meters your values will be in meters, the number 5000, could be feet or meters. Your answer would be the same .either in feet or in meters.

1. list point coordinates in sequence around the area to be calculated.
2. Cross- multiply coordinates pairs to find  Σ Northing.

(N1*E2) +(N2* E3) + (N3*E4)+.....(Nx *E1)= Σ Northing

Cross- multiply coordinates pairs to find  Σ Easting.

(E1*N2) +(E2*N3) +(E3*N4) +.....(Ex*N1) = Σ Easting

(Σ Northing ─ Σ Easting)
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬  = Area  or ( Σ N ▬ Σ E ) ÷ 2 = Area

Lets do one Acre, 43560 sq ft.

Pt 1.N 5000, E 5000
Pt.2 N 5000, E  5208.7103
Pt.3 N 5208.7103, E 5208.7103
Pt 4 N 5208.7103, E 5000
Pt.1 N5000 ,E 5000

Σ Northing Sums

N1 5000 x E2 5208.7103 = 26,043,551.5000

N2 5000 x E3 5208.7103 =26,043,551.5000

N3 5208.7103 x E4 5000 = 26,043,551.5000

N4 5208.7103 x E1 5000 = 26,043,551.5000 =Σ Northing= 104,174,206.0000

Σ Easting sums

E1 5000 x N2 5000 =25,000,000.0000

E2 5208.7103 x N3 5208.7103 =27,130,662.9893

E3 5208.7103 x N4 5208.7103 =27,130,662.9893

E4 5000 x N1 5000 =25,000,000.0000 = Σ Easting =104,261,325.9790

104,174206.0000 ▬ 104,261325.9790 = ▬87,119.9790  +/- ÷ 2 =43559.9895

43559.9895 / 43560= 1.000 Acre
 and that is how you X- multiply coordinates, to get sq ft Area or meters sq.

Now that is one way, Now i am going to use a formula that is derived by forming trapezoids and determining their area just as done in the DMD method. Meridian distances are not used. trapezoids are forum by the abscissas of the corners. Ordinates of the corners serve as the altitude of the trapezoids. Alternatively, the trapezoids can be formed by the Ordinates of the corners, and the abscissas serve as the altitudes.

Equation or formula is written like this.
Area of 12345

Area = 1/2  [ X1 x ( Y5 ▬ Y2 )
............... + X2 x ( Y1 ▬ Y3 )
................+ X3 x ( Y2 ▬ Y4 )
................+ X4 x ( Y3 ▬ Y5 )
................+ X5 x ( Y4 ▬ Y1 ) ]

Example: Traverse

pt1. N 1000.00, E 1000.00
Pt.2 N 1236.11 , E 492.03
Pt.3 N 817.72 , E 248.31
Pt 4 N 782.01 , E 622.01
Pt.5 N 574.45 ,E 854.35

Now going by my equation,

 X1,1000.00 X ( 574.45 ▬ 1236.11 ) =........................NEG, ▬ 661,660.00
 X2, 492.03 x (1000.00 ▬ 817.72 )= 89,687.2284
X3, 248.31 x ( 1236.11 ▬ 782.01) =112,757.5710
X4, 622.08 x ( 817.72 ▬ 574.45) = 151,333.4016
X5, 854.35 x ( 782.01▬ 1000.00) = ............................NEG ▬ 186,239.7565
......................................TOTAL =353,778.2010.. TOTAL +/- =847,899.7565

AREA =847,899.7565 ▬  353,778.2010 =494,121.5555/ 43560=11.3435 /2=5.6717 ACRES
same as 494,121.5555 /87,120.0000 =5.6717 Acres

The area is  equal to 1/2 of the equation as shown above in the formula.
Remember is done and formulated from DMD method. Like i said this can also be done
by forming by the Ordinates and the abscissas serve as the altitudes. this equation would look like this.

 Area of 12345
Area = 1/2 [ Y1 x ( X5 ▬ X2)
.................+Y2 x ( X1 ▬ X3)
.................+Y3 x ( X2 ▬ X4)
.................+Y4 x ( X3 ▬ X5)
.................+Y5 x ( X4 ▬ X1) ]

This equation is not that hard to Remember and it works very well.
So Now i have showed you how to find Area, 3 different ways with Coordinates.
Some still check computer or DMD's and DPD's using this method even today.
Although ,Today most Area is not calculated by hand any more , but it never hurts to know it.

What happens when the computer crashes???

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  • Hi Remus,

    Glad you like it,try my Inverse with Bearings and Area.


    Thank you for the shout back and the Thanks.

  • Hi Charlie, Like you, i have Data collectors , that will do anything. You simply do not have to calculate nothing. Just  enter the fields,and solve and it's done. and that is great. That is the way most surveying is done today ,unless you do not use one. Others rely on ,onboard Total Station programs , or lap tops or iPads.  But a Calculator in the right hands can still go out and do a survey, I can.  Calculators although from our past , can still play a important role in our work. One of the most important role it plays is with students. They have to start with it,and learn how to use it , till they get their license, after that, i hope they do not put it down to collect dust on the top of their desk. No Charlie , What you say, carries big weight with me, your knowledge and advice on anyone's discussions should be taken with great respect, from your years of experience an your knowledge of surveying. Do not take me wrong , i like you, come from the past ,i am just trying to show it , and how it was done  and that it still can work for you. Today surveyors like us live in the present,with knowledge of the past. We know two different Worlds of surveying. Today's modern surveyor of today do not, they only know how it is done today. I have pulled out a band steel chain ( steel tape) before with three crews standing there and no one knew how to help me with it , that is sad. one tried, but could not figure out how it worked, finally i got it across,thank God he did not have to use a plumb bob with it. When pulling a Chain or distance today no one calls out "CHAIN" any more. They never heard of this. So by seeing yesterdays math, no matter how elementary it is and if they like, to learn it, this gives them a little more insight on how we done it years ago. To go alone with the History of how surveyors have done their work, as those before us ,also did theirs complete different than we have done. Like you i live in both worlds. I can do and talk of these old methods, then turn around and program everything in calculators, in 4 different program  languages which i can also write and 3 on my computer. As far as State Laws, that is  a whole different subject.This must also be tackled and learned. Because this is part of the Test, and you most know these laws  to become a PLS. I will leave that to others, every State has different  Laws. So you should consult some one that has those state laws to revue them. This also goes alone with different types of systems, some states are under PLSS,while others have their own, Generally Metes and Bounds. Texas has it's own rectangular system similar to PLSS and  some of it still like, Metes and Bounds, But this only concerns  surveyors here in the USA., math goes around the World , were they have there on Grid system that has nothing to do with ours, those other countries ,have their own Laws, that have nothing to do with ours. So if i can show some old math, that someone can learn, and bring back a memory or two, I have  done what i set out to do.  No Charlie you have not done anything, but give your input, and advice,in which i value and is welcome at anytime.

    Best Regards,billy

  • Sorry, I didn't mean to steal the thread. I just thought my previous comment was misunderstood & was just trying to clarify what I was trying to say.
  • Area calculations are pretty basic. Yes, I also had the first HP calculator that had an area program. I also had the first TI calculator that had a survey chip with an area program in it. These programs are very useful & I wouldn't suggest that anyone not take advantage of them. That's not where I am coming from. They get more interesting when they involve curves. I have worked for two engineering professors & one of them taught surveying. I have also known several other professors that taught surveying classes. All of them insisted that their students learn surveying calculations the hard way BEFORE they let them use the computer. Why do you think they did that? Math, including surveying calculations, is like building a wall. First, you lay a good foundation then you build on it. If you leave out any of the key blocks, the wall is likely to fail.
    I have done pretty much every type of survey imaginable including surveys for every military base in a 3 State region, which included the survey of two military ships. Now, I am trying to be semi retired and try to restrict my efforts to boundary retracement. I try to concentrate on collecting the data I need to evaluate the situation in the field & save calculations for the office. There are always exceptions but I find that trying to do complex calculations on the hood of a pickup truck to be inefficient. The data collector on my Trimble robot will calculate most anything but I don't use it much for that so I usually have to revisit the manual. That usually wastes time. I also have a HP 50G which I rarely ever use. What I do use most is an ordinary scientific calculator with the usual trig & associated math functions. It takes longer, but I can solve any surveying problem with it if I get stuck with the need to do so in the field. The 50G will also solve the problem but I can usually solve it quicker using basic trig & math. The 50G would of course be more efficient for doing a lot of calculations, particularly if you don't have a computer. When I started my business in 1983, a TI calculator with a survey chip was all I had.
    This is all elementary stuff that most surveyors know very well. The areas that are the most problematic for surveyors, IMHO, are survey law, State plane coordinates, & geodesy. Knowing just how to manipulate the computer/calculator will get you by with elementary problems but as things get more complicated, that will become increasingly inadequate, especially when trying to resolve discrepancies/ problems.
  • Hi Everyone , Years ago , I sat in front of my drafting board , yes i said that drafting board, some times on a rainy day. Calculating Area  or Volumes, most of the time with a Planimeter , doing X- sections on road beds for Average End Area. sometimes i would do what i could on a calculator. I spend many a day with a Planimeter  with  X- section paper, that i had plotted the x-sections on .Calculating Cuts and Fills on these roads or highways. Then we had no choice, we had to do all this math,there was no other way .So miles of math has never bother me . It was just what we had to do ,to get the job done, We used a calculator, even though it was not programmable, or scientific, that we did not know ,because there was no, hand held or inexpensive  science type of a calculator.. The first HP and hand held science calculator was the HP 35s, it did not come out till 1972. They sold for $395 dollars, that was a lot of money back then. A few years later i got my first  programmable  calculator The HP 25, in 1975, although a little more time,1976, i was gave a HP 67 to work with. after that i also purchased a HP 33 E. None of these had constant memory, then years later a HP 41 CV then a CX. etc... But HP change my survey world with Programming in 1975. All that math, for End Area and Volume, and Bar Pit, i could now do on a Calculator with a program. And it still can be done today. One reason to take your test here in the U.S., you must use a Keystroke  calculator,  if programmable. These new  and older surveyors taking their test ,it is wise to have these programs,This is one reason they are still needed today. These same programs can be used in the field to day, to do one's surveying at a very efficient rate. You just have to master them.

    So in this "Area discussion" of not using a program on your calculator. He is a program for Area. This same program can be used on your Survey Exam. So here is Area By Coordinates calculator program. My Version .Hp 35s using my R<>P nested subroutine in the routine. program K. in RPN

    1. LBL K................31. RCL F...................61.RCL P

    2. CLVARS.............32.  +, plus.................62. SF 10

    3. SF10..................33.STO F...................63.PERIMETER

    4. ABC...................34.RCL E...................64. PSE

    5.PSE....................35.RCL X...................65. VIEW P

    6.NORTH................36.  -,minus,,,,,,,,,,,,,,,66. RCL F

    7.PSE....................37.RCL N...................67.  2

    8.INPUT N...............38. RCL Y..................68. /, divide

    9.STO Y..................39.  -, minus..............69. +/-, change sign

    10.STO C................40.X<>Y....................70. STO F

    11.SF 10.................41.  i,........................71. SF 10

    12.EAST.................42.  x, times..............72. AREA SQ FT

    13.PSE...................43.  +,plus................73. PSE

    14.INPUT E.............44. ARG....................74. VIEW F

    15.STO X................45. LASTx.................75. 43560

    16.STO D................46. ABS....................76. / ,divide

    17.SF 10.................47. RCL P..................77.STO A

    18.NORTH...............48.  +,plus.................78. SF10

    19.PSE...................49. STO P..................79. ACRES

    20.INPUT N.............50. RCL N..................80. PSE

    21.SF 10.................51. STO Y..................81. VIEW A

    22.EAST..................52. RCL E..................82.RTN

    23.PSE....................53. STO X

    24.INPUT E..............54. RCL D

    25.RCL Y..................55. x ± y ?, X no = Y?

    26.  x, times.............56. GTO K020

    27.RCL N.................57. RCL N

    28.RCL X..................58. RCL C

    29.  x, times.............59. x ± y ?,X no = Y ?

    30.   - ,minus.............60. GTO K020

     CK = FF7D

    LN =298

     So ,here you have it ,a program you can use at work or a program you can use at your Survey Exam. So think about how good this site is for information like this. This program cost money, and you just got it free. So help this site when you can. You can also find my Inverse with area ,here at this site, On my Discussion, The Little Big Calculator, that Calculated, I Think I Can. There is a lot of information on the HP 35s there. This inverse program, you will not find any where else, free or you have to buy it, because it is mine, It works much like this program, but it gives you all the Data around the Tract. It starts at a point and end at the same point. Just like this program but you get the Bearings and the distance also, because it is a inverse program.It will also just inverse your two points. Most only give you Azimuth, This gives you bearings. So if you like to have it go to that discussion and get it also.  I will run this program later to show you how it works.


    • Land Surveyor


      Thank you for the program, it's working great and saves time...



    •  Hi again Everyone, Like i said ,i would finish this program, as in how to use it or Run it. This is also for those of you who like to learn how to program your calculator,or just run the Program. There is a difference. But first for those of you who use Meters and see Square feet and Acres and no  Meters square or Square Meters, whatever you use. And think you can not use this.

      I can Fix that. Lets change  some of the Program and keep it simple , instead of inserting  a Choice in the program lines, that can also be done, where it will Title either one. But you probably only use one. But lets keep it simple . we could change all the variables of F, (STO F, RCL F) to STO M and RCL M, But again leave that alone, also if you like you can. So now we will change the out come and turn F into M at the End of the program. So go to line 70. STO F, that is the last time it is used. Now the next line 71. add STO M,line 72. SF 10, line 73. AREA = SQ M, line 74.PSE, line75. VIEW M, line 76. RTN, . That Ends the Program,  now you are ready for  square meters and it will say Square Meters as SQ M. Now on to how to run the Program.

      1. First , XEQ K, ENTER, you will see K001, display ABC, pse  (Area by Coordinates). NORTH

       pse. and then N ? 0.0000. on that line enter your first North.

      2. N? 0.0000, to N? 5000.0000, R/S

      3. E? 0.0000, to 5000.0000, R/S

      4. N?5000.0000 to  N? 5000.0000  since there the same, just R/S

      5.E ? 5000.0000 to E ? 5208.7103, R/S

      6.N? 5000.0000 to N? 5208.7103 ,R/S

      7.E? 5208.7103 to E 5208.7103, since they are the same, R/S,

      8. N? 5208.7103 to N ? 5208.7103, since they are the same, R/S

      9. E ? 5208.7103 to E ? 5000.0000, R/S,

      10.N? 5208.7103 to 5000.0000 ,R/S,

      11. E ? 5000.0000 to E ? 5000.000, since they are the same, R/S,

      12. then you will see PERIMETER pse, P = 834.8412, then R/S

      13. then you will see AREA = SQ M pse, M = 43599.9893, if you Run/Stop again.

       { Note :( If your using SQ FT this will say SQ FT. and F = 43599.9893,)

      Also  R/S next  ACRES pse, A = 1.0 }

      the Stack takes over and you see 834.8412 on top and 43599.9893 on the bottom of the display. This lets you know you have exited the program. If you like to run it again just press R/S and the program will initialize again, Clearing all the Data for a new calculation of a new Area. So for you who use Sq FT. used the Original Program as I wrote above, those of you who use Meters  change it like i gave instruction to do so and you will have Area BY coordinates on your HP 35s. This program  is a stand a lone program it does not require a complex

      Utilities programs. as subroutines as you will find with other large Program packs. So there is no need for it to be connect to another program.Programs like that will not work if something is wrong in the other programs. This makes it easier to trouble shoot this one.

      Sorry ,this only works in a HP35s calculator.

  • There are a number of variations of the same basic formulas. They can be handy to know at times but it is important to understand what the formula is doing, ie: how & why it works. That is an essential part of analyzing problems when they occur. I once worked with a guy in my early days as an engineering/surveying technician (early 70's) that knew how to get the answer he was looking for off of the computer. Computers were relatively crude back then but they worked & the ones we used were designed for engineers, surveyors & scientific applications & not to be a social networking toy. He usually got the right answer but had no clue what he was doing & I doubt he knew the definition a latitude or a departure. Nobody wants to do a bunch of labor intensive math calculations, let the computer do that but IMHO, knowing the principles & concepts gives one a definite advantage when solving problems. Area calculations are pretty elementary. Concepts come more into play with more complicated problems like network adjustment & post processing GPS data but knowing what the computer is doing & the math the computer program is based on is always a plus.
    • Hi Charlie , Who ever writes the program ,has to know the math. Anyone ,can load a program, and follow the steps to run it. And have not the knowledge of what it is doing. all they know the answer they want keep coming up. To them this is enough,they are satisfied and do not care to learn anymore.  The person who wrote it knows the math, Doing the math with out a program you are running one in your head, the most amazing computer ever made, not using our minds is a terrible thing to waste. And depending on something to do your work ,can leave you dependent on that device. We use to cut trees with an axe, now we use chainsaws and can cut much more than the axe, but when the chainsaw quits working, we still have the axe and it never quits. It just takes a little more time. I quest that a little like a computer and a Pencil and a piece of Paper. No matter how elementary something is if you do not know how to do it,then it  is not simple then, without this elementary knowledge it becomes unsolvable  and complex.

       Today modern surveying and engineering very seldom resorts to doing hand calculations, only as a check in some cases.  When some one there knows how to do it. I have seen wrong Area put on Documents, At a embarrassing  date later to be changed, At that time they wished somebody had check it.  But this is for those, who have never seen it and would like to learn how, and for those with memories of when they did it, to look back into the Past. It was never intended to be a Rocket Science. But i bet there is a lot who think, how does this computer know how many square ft. is in this survey. Well now they know, in around about way.

      Have a Good Day Charlie, and Thanks for the Reply.

      • Billy, thanks for the memories !

        Back in the Seventies, one of my task (on a road job) was to calculate cross sectional area at every Station using the method you mentioned above, then the volume (earth cut/fill). My bugbear was Pappus theorem for curved portion of road, it was later agreed to omit this theorem to simplify life.

        I agree that with computer, thing was much easier, but not understanding the principle behind it, makes one just a data entry person.


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