Analytical method for calculating the surface area of the polar coordinates
Calculate the surface area of the polar coordinates is dependent on the number of positions, from which measurements were made.
⇒ For one position to designate the area to be on the ground from his position to make a minimum of two points with known coordinates, measure the length and directions to points of the contour bends. Then you need to make a reduction of measured directions*.
*If you do not remember how to reduce the measured directions, click
Figure 1. The siting of points 123 in the collapse of a contour in coordinate system
Search:
P
Measured
length: d_{1}, d_{2}, d_{3}
Directions reduced: k_{1}, k_{2}, k_{3}
Field P the polygon 123 calculate from formula:
2P = ∑_{1}^{n } d_{i}d_{i+1}sin(k_{i+1} – k_{i})
where:
n – number of contour points of collapse,
i – the number of point.
The control is to determine whether:
∑_{1}^{n } (k_{i+1} – k_{i}) = 0
Field measurements are given in hectares, remembering that 1ha = 100a = 10 000m^{2}.
The following is a table that facilitates the calculation of surface area.
Point number 
k_{i} 
d_{i} 
d_{i}d_{i+1} 
k_{i+1} – k_{i} 
1 
k_{1} 
d_{1} 
d_{1}d_{2} 
k_{2} – k_{1} 
2 
k_{2} 
d_{2} 
d_{2}d_{3} 
k_{3} – k_{2} 
3 
k_{3} 
d_{3} 
d_{3}d_{1} 
k_{1} – k_{3} 
1 
k_{1} 
d_{1} 
_ 
∑_{1}^{n } (k_{i+1} – k_{i}) 
⇒ If the lengths and directions to points of the contour collapse were measured from several observation stations (Figure 2) we need to know the coordinates of the observation stations. Then we calculate coordinates of a collapse contour points:
X_{i} = X_{S} + d_{i}cosA_{Si}
Y_{i} = Y_{S} + d_{i}sinA_{Si}
Continuation of the calculations performed of the Gauss's formulae*.
*If you do not remember how to calculate the surface area of the Gauss's formulae, click
Figure 2. Location points 123456 in the collapse of a contour in coordinate system
Note: To be able to calculate the surface area of the polar coordinates measured from several observation stations, the observation stations must be in one coordinate system.
