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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???

This Content Originally Published by a land surveyor to Land Surveyors United Network

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Thanks for that. Love this site and always little tips that go a long way. I can't count the number of times the battery or laptop dies on site or I'm a km into the bush.

Those r helpful formulas. Gud job surv. Billy

 Hi Everyone, I like to remind or tell anyone new to doing this, any Area calculations, using Lat. and Dep. in DMD or DPD or Area by Coordinates,  These values have to be adjusted before you make you calculations, Lat. and Dep, must be balance, in DMD or DPD and coordinates  also must be adjusted, any method you use, this must be done  before the correct Area can be calculated, It must Close. You choose what method you like, to adjust with. Compass Rule is find.

Any thing else is just a rough calculation of that Area.

Thank you, ED, and thank you Adeulure, This in away should or maybe been in Once Upon Time Discussion. But Like all of it it still can be used today. I said i would show it there, but got involved there.with going over how to calculate a simple curve using old method. i just attached a file on calculating Deflection Angles,two different ways ,with just your calculator, and with the Short chords, very good reading. Go there and check it out. All it takes is you and your calculator. Survey Students need to learn all of these methods.  Some have never even tried to do DMD, DPD, and using Coordinates, maybe X- multiplying , All these old equations and formulas ,where taught to me ,by my old mentors one of which was, William E. Creel, PLS Who started his Surveying in 1949. He taught me many things,one of which is

Irregular Areas, Cutting Off, with just a calculator. Him and I  have shared ,many a Survey adventure. And no matter who i went to work for,His door was always open for me, if i needed work.  But he was not the only one, Jack Duccan , PE/ PLS was older than Mr. Creel,he also shared his knowledge  and there where many more. So i like to past all this on.One of the oldest survey spirits , is sharing with other surveyors. Like this site, it is not just a chat room.

and thank both of you again for your reply's.

Hi again Everyone, This is for those who may not use a programmable calculator. students manly. But there are calculator programs to do this. Some are downloadable others are on Cards or Module's, and are very handy to use. The programmable calculators allow on a test ,you have to program your self in keystroke programming, and there is always the chance ,that you could lose memory in a test. So far as one of my friends took 3 calculators, same programs to the test. But if you did not have that option, these would come in handy. I have many keystroke programs that can do this, one is Inverse with Area,that i shared here at LSU, then there is Area by Coordinates,  then a Traverse with Area, and many more. If you do not have this option, then this is your solution.This is a circle, there are things in the past that can help you, as well as new things in the present. These are some visual aids, for Cross x Multiplying  Coordinates ,just thought i would add them. In  Coordinate Geometry Fundamentals  pages 42,43,44 and there is a lot of good stuff here also all this can be done on just a calculator, no programs.  Then from engineeringsurveyor.com  a GIF that also shows this method. This will help you with mine, on Cross Multiplying Coordinates, this is used manly to day for checking Area.



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.


ALFRED, Hi, and you are very kind. Computers and Calculators are my life. Thanks for your Reply. Look forward to here from you again.

Best Regards

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.


 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.


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




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