In triangulation there are three types of triangulation networks (or nets). They are the chain of single and chain of quadri-laterals.

The simplest triangulation system is the chain of single triangles shown in figure 15-15. Suppose AB is A (that is, the observed angle BAC) measures 9854 and that angle ABC measures ACB.

Next, solve for sides BC and AC by using the law  

Now that you know how to find the length of BC, you can proceed in the same manner to determine the lengths of BD and CD. Knowing the length of CD, you can proceed in the same manner to determine the lengths of CE and DE, knowing the length of DE, you can determine the lengths of DF and EF, and so on. You should use this method only when locating inaccessible points, not when a side of the triangle is to be used to extend control.

In comparison with the other systems about to be described, the chain of single triangles has two disadvantages. In the first place, it can be used to cover only a relatively narrow area. In the second place, it provides no means for cross-checking computed distances using computations made by a different route. In figure 15-15, for example, the only way to compute the length of BC is by solving the triangle ABC, the only way to compute the length of CD is by solving the triangle BCD (using the length of BC previously computed); and so on. In the systems about to be described, a distance maybe computed by solving more than one series of triangles.

Technically speaking, of course, a triangle is a polygon; and therefore a chain of single triangles could be called a chain of polygons. However, in reference to triangulation figures, the term chain of polygons refers to a system in which a number of adjacent triangles are combined to forma polygon, as shown in figure 15-16. Within each polygon the common vertex of the triangles that compose it is an observed triangulation station (which is not the case in the chain of quadrilaterals described later).

You can see how the length of any line shown can be computed by two different routes. Assume that AB is EF. You can compute this length by solving

triangles ADB, ADC, CDE, and EDF, in that order, or by solving triangles ADB, BDF, and FDE, in that order. You can also see that this system can be used to cover a wide territory. It can cover an area extending up to approximately 25,000 yards in length or breadth.