Calculating Included Angles in Surveying

In surveying, understanding how to calculate included angles from bearings is fundamental for traverse calculations. The included angle at a station is the angle between two lines meeting at that station. It is usually measured clockwise from the preceding line to the succeeding line.

Understanding Bearings

• Fore Bearing (FB): The direction of a line measured clockwise from the North meridian at the starting point of the line.
• Back Bearing (BB): The direction of a line measured clockwise from the North meridian at the ending point of the line. The Back Bearing of a line differs from its Fore Bearing by exactly $180°$.

Given:

• Fore bearing of line AB = $40°$
• Back bearing of line BC = $280°$

We need to find the included angle ABC. This is the angle at station B between the lines BA and BC.

Step-by-Step Calculation

To find the included angle at B (ABC), we need the bearings of the lines meeting at B, which are line BA and line BC. We are given the fore bearing of AB and the back bearing of BC.

1. Calculate the Bearing of Line BA

The line BA is the preceding line to BC at station B. Its bearing is the back bearing of line AB.

Back Bearing of AB = Fore Bearing of AB $\pm 180°$.

Since the Fore Bearing of AB ($40°$) is less than $180°$, we add $180°$.

Bearing of BA = $40° + 180° = 220°$.

So, the bearing of line BA is $220°$.

2. Calculate the Bearing of Line BC

The line BC is the succeeding line at station B. We are given the back bearing of line BC is $280°$. The bearing of line BC is its fore bearing.

Fore Bearing of BC = Back Bearing of BC $\pm 180°$.

Since the Back Bearing of BC ($280°$) is greater than $180°$, we subtract $180°$.

Bearing of BC = $280° - 180° = 100°$.

So, the bearing of line BC is $100°$.

3. Calculate the Included Angle ABC

The included angle at B (ABC) is the angle between the line BA and the line BC, measured clockwise from BA to BC.

We have the bearing of BA as $220°$ and the bearing of BC as $100°$. Both bearings are measured clockwise from the North direction at point B.

Imagine a compass at point B. Line BA is at $220°$ from North. Line BC is at $100°$ from North.

To get from the direction of BA ($220°$) to the direction of BC ($100°$) by moving clockwise, we must pass through the North direction ($360°$ or $0°$).

The angle from the direction of BA to the North direction (clockwise) is $360° - 220° = 140°$.

The angle from the North direction to the direction of BC (clockwise) is $100°$.

The total included angle ABC is the sum of these two angles:

Included Angle ABC = (Angle from BA to North) + (Angle from North to BC)

Included Angle ABC = $140° + 100° = 240°$.

Alternatively, a common formula for the included angle at a station (measured clockwise) is:

Included Angle = Bearing of Succeeding Line - Bearing of Preceding Line $\pm 360°$.

Here, the succeeding line is BC (Bearing $100°$) and the preceding line is BA (Bearing $220°$).

Included Angle ABC = Bearing of BC - Bearing of BA

Included Angle ABC = $100° - 220° = -120°$.

Since included angles are typically positive and less than $360°$, a negative result means the angle is $360° - | \text{result} |$.

Included Angle ABC = $360° - 120° = 240°$.

Both methods yield the same result.

The included angle ABC is $240°$.

Calculation of Included Angle ABC:

Bearing of BA = $220°$
Bearing of BC = $100°$

Included Angle ABC = (Bearing of BC - Bearing of BA) if result is positive, otherwise add 360° OR Angle clockwise from BA to BC.

From BA ($220°$) to BC ($100°$) clockwise:

$(360° - 220°) + 100° = 140° + 100° = 240°$

Using formula: $100° - 220° = -120°$. Add $360°$ to get positive angle: $-120° + 360° = 240°$.

Revision Table: Key Concepts

Additional Information on Surveying Angles

Included angles are crucial for checking the accuracy of a closed traverse. The sum of the interior angles in a closed traverse with $n$ sides should equal $(2n - 4) \times 90°$. Exterior angles are also sometimes measured, and their sum is $(2n + 4) \times 90°$. When calculating included angles from bearings, it's important to be consistent with the direction of measurement (clockwise or counter-clockwise) and whether you are calculating interior or exterior angles. The method used above calculates the interior angle measured clockwise from the back end of the preceding line (BA) to the front end of the succeeding line (BC).