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Example
Trapezoidal Rule
The area AXYZCBA is typical of part of a rural allotment bounded on one side by an irregular side (eg. a creek). The regular part of the allotment has been excluded from these calculations by the use of the traverse line ABC. The area below this line can be computed by means of triangles as shown in part 1 (and other methods shown later). All that remains is to compute the irregular area AXYZCBA. This is done by approximating the area by a series of equally spaced trapezia, measuring these either in the field or off a plan, and then computing the area of each of these.
Using the rules of Euclidean Geometry, the area of the first trapezoid is given by;
| A1 = | L(O1 + O2) |
| 2 |
where L is the constant distance along the traverse line between offsets O1 and O2. Now the total area of the figure is given by;
AT = A1 + A2 + A3 + A4 + A5
so substituting the particular elements, in terms of On and L, the total area is given by;
AT =L[(O1 + O2) + (O2 + O3) + (O3 + O4) + (O4 + O5) + (O5 + O6)]/2
| AT = | L(O1 + O6) | (O2 + O3 + O5 + O5) |
| 2 |
and in more general terms the Trapezoidal Rule may be quoted as;
| A = | L(O1 + On) | (O2 + O3 + O4 + On-1) |
| 2 |
| or A = | L | (O1 + On + 2SOthers) |
| 2 |
(1st + last + 2 times the sum of the others)
where O1 .. On are offsets; L is the uniform distance between offsets and n may be odd or even. The resulting area is generally less than the true area. The accuracy of the area will depend on the number of offsets (and therefore the distance between them) and the degree of irregularity of the boundary. Of course the more irregular the boundary the more offsets should measured; this will demand a compromise between the time spent gathering the data and the required accuracy.