There are certain checks adopted for traversing the angular work. In this regard, the sum of all the exterior angles of a closed traverse having six sides is equal to
1440°
Calculating Angular Work in a Closed Traverse
Traversing is a fundamental surveying method used to determine the relative positions of points. A closed traverse starts and ends at the same point, forming a polygon. Angular measurements are critical in traversing, and checks are applied to ensure accuracy. One such check involves the sum of the angles of the traverse.
Understanding Angles in a Closed Traverse
For any closed polygon or traverse with $n$ sides, there are two main types of angles measured at each vertex:
- Interior Angles: These are the angles measured inside the polygon. The sum of the interior angles of a closed traverse with $n$ sides is theoretically equal to $(n-2) \times 180^\circ$.
- Exterior Angles (Standard Definition): These are the angles formed by extending one side of the polygon and measuring the angle between the extended side and the next side. The sum of the exterior angles (one at each vertex) of any convex polygon, regardless of the number of sides, is always $360^\circ$.
Interpreting "Exterior Angles" for this Question
The question asks for the sum of exterior angles of a closed traverse with six sides. Given the provided options and the correct answer, it appears the question might be defining "exterior angle" differently than the standard geometric definition. A common interpretation in some contexts, when dealing with traverse closure, is to consider the 'exterior' angle at a vertex as the reflex angle, or $360^\circ$ minus the interior angle.
Let's proceed with this interpretation to match the provided answer.
Calculating the Sum of Interior Angles
The traverse has six sides, so the number of sides $n = 6$.
The theoretical sum of the interior angles of a closed traverse with $n$ sides is given by the formula:
\(\text{Sum of Interior Angles} = (n-2) \times 180^\circ\)
Substituting $n=6$:
\(\text{Sum of Interior Angles} = (6-2) \times 180^\circ\)
\(\text{Sum of Interior Angles} = 4 \times 180^\circ\)
\(\text{Sum of Interior Angles} = 720^\circ\)
Calculating the Sum of "Exterior Angles" (as 360° - Interior Angle)
If we define the 'exterior angle' at each vertex as \(360^\circ\) minus the interior angle at that vertex, then the sum of these 'exterior angles' for the entire traverse would be the sum of \((360^\circ - \text{Interior Angle})\) for all vertices.
For a traverse with $n$ vertices (and $n$ sides), the sum would be:
\(\text{Sum of 'Exterior' Angles} = \sum_{i=1}^{n} (360^\circ - \text{Interior Angle}_i)\)
This can be rewritten as:
\(\text{Sum of 'Exterior' Angles} = n \times 360^\circ - \sum_{i=1}^{n} \text{Interior Angle}_i\)
\(\text{Sum of 'Exterior' Angles} = n \times 360^\circ - (\text{Sum of all Interior Angles})\)
Using $n=6$ and the calculated sum of interior angles ($720^\circ$):
\(\text{Sum of 'Exterior' Angles} = 6 \times 360^\circ - 720^\circ\)
\(\text{Sum of 'Exterior' Angles} = 2160^\circ - 720^\circ\)
\(\text{Sum of 'Exterior' Angles} = 1440^\circ\)
Following this interpretation of 'exterior angle', the sum for a six-sided closed traverse is $1440^\circ$. This calculation aligns with option 1.
| Number of Sides (n) | Sum of Interior Angles | Sum of 'Exterior' Angles (assuming \(360^\circ\) - Interior Angle) |
|---|---|---|
| 3 | \((3-2) \times 180^\circ = 180^\circ\) | \(3 \times 360^\circ - 180^\circ = 1080^\circ - 180^\circ = 900^\circ\) |
| 4 | \((4-2) \times 180^\circ = 360^\circ\) | \(4 \times 360^\circ - 360^\circ = 1440^\circ - 360^\circ = 1080^\circ\) |
| 5 | \((5-2) \times 180^\circ = 540^\circ\) | \(5 \times 360^\circ - 540^\circ = 1800^\circ - 540^\circ = 1260^\circ\) |
| 6 | \((6-2) \times 180^\circ = 720^\circ\) | \(6 \times 360^\circ - 720^\circ = 2160^\circ - 720^\circ = 1440^\circ\) |
This table illustrates the pattern for different numbers of sides and confirms our calculation for $n=6$. It is important to note that the standard geometric definition of exterior angles gives a sum of $360^\circ$ for any polygon. However, the context of traverse checks might imply a different definition, leading to the result $1440^\circ$ for a six-sided traverse.
Revision Table: Key Traverse Angle Concepts
| Concept | Definition | Formula for Sum (n-sided closed traverse) |
|---|---|---|
| Interior Angle | Angle inside the polygon at a vertex. | \((n-2) \times 180^\circ\) |
| Exterior Angle (Standard Geometric) | Angle between one side and the extension of the adjacent side. | \(360^\circ\) (for any convex polygon) |
| "Exterior Angle" (as $360^\circ$ - Interior Angle) | Reflex angle at a vertex, or \(360^\circ\) minus the interior angle. | \(n \times 360^\circ - (n-2) \times 180^\circ = (n+2) \times 180^\circ\) |
| Deflection Angle | Angle between the extension of the previous line and the next line. | Sum is \(360^\circ\) (considering left and right deflections with sign) |
Additional Information: Angular Checks in Traversing
Angular checks are vital for assessing the accuracy of angular measurements in a traverse before proceeding with linear measurements and coordinate calculations. The sum of the measured angles in a closed traverse should theoretically match the calculated theoretical sum based on the number of sides. The difference between the observed sum and the theoretical sum is called the angular misclosure.
- For interior angles, the theoretical sum is \((n-2) \times 180^\circ\). Angular misclosure is \(\sum \text{Measured Interior Angles} - (n-2) \times 180^\circ\).
- For deflection angles, the theoretical sum is \(360^\circ\) (algebraic sum, with right deflections typically positive and left deflections negative). Angular misclosure is \(\sum \text{Measured Deflection Angles} - 360^\circ\).
The acceptable limit for angular misclosure depends on the desired precision of the survey. If the misclosure is within the acceptable limit, it is distributed among the measured angles before further calculations are performed.
Important Questions from Traverse Surveying
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If the perimeter of traverse is 2000 m and the amount of closing error is 10 m, then the relative closing error would be:
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If the reduced bearing of a line AB is N60° W and the length is 100 m, then the latitude and departure of the line AB will be,
