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Triangulation, trilateration; specifications

Short answer

Triangulation measures one baseline and all angles to compute other sides by the sine rule. Trilateration measures all sides with an EDM and computes angles by the cosine rule. First‑order triangulation: sides 30–150 km, baseline 5–15 km, average triangle closure <1″. Second‑order: sides 8–30 km, baseline 1.5–5 km, closure 3″. Third‑order: sides 1.5–10 km, baseline 0.5–3 km, closure 6″.

What is triangulation and trilateration? Write down the general specifications of the different orders (types) of triangulation.

Triangulation is a method of horizontal control survey in which control stations are fixed at the vertices of a chain or net of triangles. One side (the base line) is measured very accurately, all the angles of the triangles are measured, and the lengths of the other sides are computed with the sine rule.

Trilateration is the method in which the lengths of all the sides of the triangles are measured directly with an EDM (electromagnetic distance measuring) instrument, and the angles are computed from the sides using the cosine rule.

Principle

The base line AB is measured accurately. At A and B the angles to a third station C are measured. In triangle ABC, the sides AC and BC are computed by the sine rule:

ABsin⁡C=BCsin⁡A=ACsin⁡B\frac{AB}{\sin C} = \frac{BC}{\sin A} = \frac{AC}{\sin B}

The computed side BC (or AC) is then the base for the next triangle BCD, and so on. The whole area is covered by a chain or net of triangles. At the end, a second base line (check base) is measured and compared with its computed length; the difference shows the accuracy of the work. Azimuth is observed at the ends to control the direction.

      C-----E-----G
     / \   / \   / \
    /   \ /   \ /   \
   A-----B-----D-----F
   base                   check base

General specifications of the different orders of triangulation

ItemFirst order (primary)Second order (secondary)Third order (tertiary)
Length of base line5 to 15 km1.5 - 5 km0.5 - 3 km
Length of sides of triangles30 - 150 km8 - 30 km1.5 - 10 km
Average triangular error (closure)less than 1"3"6"
Maximum triangular error3"8"12"
Probable error of base line1 in 1,000,0001 in 500,0001 in 250,000
Discrepancy between two measures of the base5 mm K\sqrt{K}10 mm K\sqrt{K}20 mm K\sqrt{K}
Probable error of computed distances1 in 60,000 to 1 in 250,0001 in 20,000 to 1 in 50,0001 in 5,000 to 1 in 20,000

(KK is the distance in km. The figures are the common textbook values; Survey Department practice may differ slightly.)

Primary (first order) triangulation gives the main control of a country, secondary triangulation fills in the primary net, and tertiary triangulation provides closely spaced stations for engineering and topographic surveys.

✓ Answer checked by Claude Opus 5.5 (high effort)

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