Chain Surveying and Triangle Plotting Accuracy

Chain surveying relies on forming a network of triangles over the area to be surveyed. The accuracy of the final map or plan depends significantly on the shape of these triangles. Triangles are used because their shape is uniquely determined by the lengths of their sides, which are measured during the survey.

Why Triangle Shape Matters in Surveying

When plotting the measured side lengths on paper, the intersection points of the arcs drawn from the ends of a baseline determine the third vertex of the triangle. If the angles of the triangle are either very acute (close to 0 degrees) or very obtuse (close to 180 degrees), a small error in measuring a side length or in drawing the arc during plotting can cause a large displacement in the position of the third vertex. This leads to inaccuracies in the survey.

Triangles with angles that are neither too small nor too large are considered "well-conditioned" and lead to more accurate plotting. The ideal range for angles in surveying triangles is generally between 30 degrees and 120 degrees.

Analyzing Triangle Types for Accuracy in Chain Surveying

Let's look at the types of triangles mentioned and their suitability for accurate plotting in chain surveying:

• Obtuse angled triangle: This type of triangle has one angle greater than 90 degrees. If the obtuse angle is very large (close to 180 degrees), the opposite side is very long, and the other two angles are very acute (close to 0 degrees). Plotting such a triangle is highly susceptible to errors as the arcs intersect at a very sharp angle, making the intersection point difficult to determine precisely.
• Right angled triangle: This triangle has one angle equal to 90 degrees. While better than very acute or obtuse triangles, a right angle (90 degrees) is not the most ideal for accuracy. Angles closer to 60 degrees are generally preferred.
• Isosceles triangle: An isosceles triangle has two equal sides and two equal angles. While some isosceles triangles can have well-conditioned angles (e.g., an isosceles triangle with angles 50°, 65°, 65°), others might have angles that are too acute or too obtuse (e.g., an isosceles triangle with angles 20°, 80°, 80° or 140°, 20°, 20°). Its suitability depends on the specific angles.
• Equilateral triangle: An equilateral triangle has all three sides equal in length and all three angles equal to 60 degrees ($\frac{180^\circ}{3} = 60^\circ$). An angle of 60 degrees is considered the ideal angle for triangulation in surveying because it falls perfectly within the optimal range (30° to 120°) and provides the most definite intersection point when plotting. Triangles with angles near 60 degrees minimize the effect of small errors in measurement and plotting on the final position of points.

Conclusion

Based on the principles of triangulation and the effect of angle size on plotting accuracy, a triangle with angles close to 60 degrees is the most desirable shape in chain surveying. The equilateral triangle, having all angles exactly 60 degrees, represents the most accurately plottable triangle type among the options provided.

Therefore, an equilateral triangle is considered the most accurately plotted in chain surveying because its angles ($\ang{60}$) provide the strongest intersection for vertex determination, minimizing the impact of measurement or plotting errors.