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
                        2


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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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.
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.

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

Hi Remus,

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

and 

Thank you for the shout back and the Thanks.

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