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 ) ]
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???