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# AREA by Coordinates, on a Calculator,Without a Program

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

Views: 722

### Replies to This Discussion

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.

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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