G. Horizontal Angles

1. Definition

A horizontal angle is measured perpendicular to gravity. Three points are involved:

  • Backsight (BS)

  • Foresight; FS

  • At


The angle is between two vertical planes; one contains the BS and At points, the other contains the FS and At. The At point is generally occupied by a surveying instrument, Figure G-1.


Figure G-1
Horizontal Angle

If we look down on the horizontal plane with the three points, Figure G-2, we see there are two horizontal angles between the vertical planes: HAng1 and HAng2.


Figure G-2
Two Horizontal Angles

This causes an ambiguous situation unless we can specifically identify which angle we mean. Recall that the four parts of an angle are:

  • Start

  • Direction

  • Magnitude

  • End


HAng1 and HAng2 have the same start (BS) and end (FS) but different direction and magnitude. All four parts must be clearly defined. Merely including magnitude without a direction doesn’t resolve the ambiguity. For example, a 135° angle from BS to FS could still be interpreted two ways, Figure G-3.



Figure G-3
Magnitude without Direction

Similarly, expressing a direction, to the right (clockwise) or left (counterclockwise), without a magnitude is also unclear, Figure G-4.


Figure G-4
Direction without Magnitude

There are a few different ways to specify horizontal angles which help minimize confusing interpretations.

2. Types of Horizontal Angles

a. Interior/Exterior

On a closed non-crossing polygon, Figure G-5, horizontal angles can be either interior (red) or exterior (blue).

Figure G-5
Interior and Exterior Angles


The geometric condition for a non-crossing polygon is Equation (G-1).

h ang 04e

Equation (G-1)

Interior and exterior don’t make sense if the polygon is open, Figure G-6(a), or if the polygon crosses itself, Figure G-6(b).


a. Open Polygon

b. Crossing Polygon

Figure G-6


In the latter case, the polygon turns itself inside-out and Equation (E-1) does not apply. Open and crossing polygons need some other way to express an angle.


b. Right/Left

Right or left is the rotational direction from the BS point to the FS point. Consider standing on the At point, looking at the BS point: an angle to the right means you physically turn your body to the right to see the FS point. An angle right is clockwise, left counter-clockwise, Figure G-7.


Qt SVG Document

Figure G-7

Figure G-8(a) shows a closed non-crossing polygon with interior angles to the right; Figure G-8(b) to the left.


a. Angles Right

b. Angles Left

Figure G-8
Interior Angles

The dashed lines in Figure G-8 show the angle measurement sequence. In Figure G-8(a), interior angles to the right measurement progresses counter-clockwise around the polygon.


We can also have exterior angles to the right, Figure G-9(a), and to the left, Figure G-9(b).


a. Angles Right

b. Angles Left

Figure G-9
Exterior Angles


Right or left help with angle definitions on the open and crossing polygons. In Figure G-10, progressing in the direction of the dashed line, the open polygon has angles to the right.


Figure G-10
Angles Right on Open Polygon


Figure G-11 shows a crossing polygon using angles left progressing in the direction of the dashed line.



Figure G-11
Angles Right on Crossing Polygon

Note how the angles change from “interior” to “exterior” although there is no consistent polygon interior.

c. Deflection Angles

A deflection angle is how much the next line deflects from an extension of the previous line. It consists of two parts: (1) magnitude and (2) direction. For example, in Figure G-12:

Line BC deflects from line AB 45° to the right, written as 45°R.

Line CD deflects from line BC 30°L.



Figure G-12
Angles Right on Open Polygon


A deflection angle ranges from 0° (no deflection) to 180° (going back along the preceding line).


The mathematical angle condition for a closed non-crossing polygon, Figure G-13, is:


Equation (G-2)

Right deflection angles are positive (+), left are negative (-).


Figure G-13
Deflection Angles Around Polygon

Equation (G-2) is also the mathematical condition for a closed polygon which crosses itself an even number of times. For an odd number of crossings, Figure G-14, the mathematical condition is:


Equation (G-3)


Figure G-14
Odd Crossings Polygon

How does reversing travel direction around a polygon change the deflection angles? Figure G-15 shows that the deflection angle value is the same, but its direction is reversed.


a. Clockwise Travel

b. Counterclockwise Travel

Figure G-15
Reversing Travel Direction

3. Example Problem

Given this polygon with combination of angle types:



a. Deflection angle at A from D to B

Defl angle A = 180°00’ - 113°20’ = 66°40’ = 66°40’ R



b. Angle left at B from A to C


Int angle B = 180°00’ - 92°42’ = 87°18’


c. Deflection angle at D from A to C



Need interior angle at D to compute the deflection angle.


To compute interior angle at D, subtract the sum of the other interior angles from (n-2)x180°.

Need interior angle at C.



Int angle C = 360°00’ - 259°35’ = 100°25’



Int angle D = (4-2)x180°00’ - (113°20’ + 87°18’ + 100°25’) = 360°00’ - 301°03’ = 58°57’




Defl angle D = 180°00’ - 58°57 = 121°03’ = 120°03’ L