The document discusses angular measurement in surveying. It describes the theodolite instrument used to measure horizontal and vertical angles. It defines important terms like centering, vertical axis, horizontal axis, face left/right, plunging, and line of collimation. It explains how to set up, center and level a theodolite. It also discusses measuring horizontal angles, vertical/zenith angles, interior angles, deflection angles, and determining the direction and bearing of a line from its azimuth.

1. Addis Ababa University Ethiopian Institute Of Architecture, Building
Construction And City Development
Compiled by Ebisa Tesfaye (Msc.) Page 1
CHAPTER 4
Measurement of Angle
4.1 Introduction
In surveying the direction of a line is described by the horizontal angle that it makes with a
reference line or direction.
4.2 Instrument for angle measurement
Theodolite is the instrument used to perform accurate angular measurements in surveying. It is
used for measuring horizontal and vertical angle.
4.2.1 Terms used in angular measurements
The following terms should be clearly understood.
1. Centering: it is the process of setting up the instrument exactly over the station mark.
2. Vertical axis: the vertical axis of the theodolite is the axis about which the instrument rotates
in the horizontal plan.
3. Horizontal axis: it is the line passing through the centers of the journals, which fit into the
bearings at the top of the standards.
4. Face left: if the vertical circle is on the left side of the observer, the theodolite is in the face
left.
5. Face right: if the vertical circle is on the right of the observer, the theodolite is in the face
right.
6. Plunging the telescope: it is the process of rotating the telescope over the horizontal axis
through 1800
in the vertical plane. Plunging is also known as transiting or reversing.
7. Swinging the telescope: is the process of turning the telescope about the vertical axis in a
horizontal plane.
8. Changing face: the telescope is transited and then followed by 180°turn in the horizontal
plane.
9. Line of collimation (line of sight): this line is defined by the cross hairs & the optical centre
of the objective.
4.2.2 Axis of the theodolite
- The collimation axis ZZ should be normal to the horizontal
axis KK!
- The horizontal axis KK should be normal to the vertical axis
VV!
- The plate bubble axis LL should be horizontal!!
- The vertical axis VV should be vertical (in the plumb line!)
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4.2.3 Parts of the theodolite
1. Hand grip:- used to handle the instrument
2. Peep sight:- used to target the point to be measured
3. Vertical tangent screw:- used to fasten the telescope on vertical plane
4. Eyepiece: - used to read the angle
5. Vertical motion screw: - used to rotate the telescope on vertical plane slowly
6. Horizontal tangent screw: - used to fasten the telescope on horizontal plane
7. Tribrach: - is the lower part of the theodolite
8. Optical plummet: - used to see the ground point (station mark)
9. Focusing screw: - used to sharp and clear the cross hair
10. Microscope eyepiece: - used for reading the scale
11. Horizontal motion screw: - used to rotate the telescope on horizontal plane slowly
12. Circular bubble and Plate bubble tube:- is used to adjust the instrument in a horizontal
plane.
4.2.4 Preparing a theodolite for work
Preparing a theodolite for work is required to be made at each setting of the instrument
before taking observations.
To make ready the theodolite for work the following five steps are required
A. Setting up the theodolite
B. Centering the theodolite
C. Leveling up the theodolite
D. Focusing the eyepiece of the theodlite
E. Focusing the objective of the theodlite
Steps for setting up, centring and leveling a theodolite by means of an optical plummet:-
1. Extend the tripod legs to suited length (for observer) should sight through the telescope
comfortably & setting up the tripod approximately over the station (the ground point), by
placing the leg at equal distance from ground point.
2. Clamp (attach) the theodolite to the tripod head by means of the fastening screw
3. Looking through the optical plummet, centre the cross hair on the survey point by fixing
one leg & moving the two legs the same amount on the ground.
4. Now the optical plummet is cantered but the circular bubble is off centre. Loosen the
clamps on the tripod leg which is in the direction you want the bubble to move ( to bring) in
to centre. Adjust the legs turn by turn until the bubble moves towards the ring by shortening
or extending it.
5. Check the survey point by the optical plummet. If it’s out of the cross hair, by loosening
the fastening screw, slide the body of the instrument on the tripod head. Check the bubble.
6. Rotate the telescope until the cylindrical bubble (plate bubble) is aligned with any two
foot screws.
7. Centre the bubble by using the two foot screws 1 and 2 by turning in opposite direction.
8. Next Rotate the instrument (900
) or until the cylindrical bubble is aligned to the 3rd
foot
screw
9. Centre the bubble using the 3rd
foot screw.
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10. Rotate the telescope 1800
or 3600
, if re- centring is required; repeat the procedure until it
remains centre.
4.3 Measurement of angle and direction
In surveying angles are measured either in a horizontal plane yielding horizontal angles or in
vertical plane yielding vertical angles.
4.3.1 Horizontal angle
In figure 1 points A,B,C are three points located on the earth’s surface points A’ ,B’,C’ are the
projection of points A,B, and C onto a horizontal plane . angles A’B’C’, B’C’A’ & C’A’B’ are
the horizontal angles.
4.3.2 Vertical and Zenith angles
A vertical angle is measured in a vertical plane using a horizontal plane as reference plane.
When the point being sighted is above the horizontal plane the vertical angles is called an angle
of elevation & is considered as a positive angle. When the point being sighted is below the
horizontal plane, the angle is called an angle of depression and considered as a negative angle.
The value of vertical angle can range from -900
to +900
(-100 grad to+100 grad).
A Zenith angle is also measured in a vertical plane but uses the overhead extension of plumb line
as a reference line. Its value ranges from O0
to 1800
(0 grad to 200 grad). In figure 2 the zenith
angles measured at station A to targets at station B & C are 700
& 1200
. The corresponding
vertical angles are 200
and -300
.
Most of the theodolite gives zenith angles.
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Angles are usually observed twice by changing face of the instrument (face right reading and
face left reading). The average of the two observations is taken as the value of the angle.
Interior angles
The angles formed within a closed figure b/n adjacent sides are known as interior angles the sum
of interior angles of a polygon of n sides equals : (n-2)* 180
Based on this value one can check the accuracy of field work (measurements).
Deflection angle
The angle measured from the prolongation of the preceding line to the following line is called
deflection angle. Deflection angles are frequently measured in surveys for high ways, railways,
and transmission lines and so on. Deflection angles may have values between 0 and 180 degrees.
In any closed polygon
The algebraic sum of the deflection angles for closed figure, considering right deflections as
positive and left deflections as negative is 3600
.
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The direction of a line
Direction of a line is defined as a horizontal angle that it makes either with adjacent line of
survey or with some real or imaginary reference line of fixed direction called MERIDIAN.
There are three types of meridians:-
True meridian: a reference line that passes through the geographic north and south poles is
called true meridian.
Arbitrary (Assumed) Meridian: is a meridian (reference line) chosen arbitrarily. Any line can
be chosen as an arbitrary meridian.
Magnetic meridian: the direction (line) defined by a freely suspended, balanced, magnetic
needle that lies parallel to the magnetic lines of force of the earth is called the magnetic meridian.
Bearings and Azimuths
Azimuth of a line is the horizontal angle measured on the ground usually from North in clock
wise direction up to a given line in the range of 00
to 3600
. Azimuth is also called whole circle
bearing (WCB).
In the above diagram, Az. AB=600
, Az. AC=1400
, Az. AD=2100
Back Azimuth
AZAB=700
Back azimuth of AB= Az. of BA=700
+1800
=2500
The back azimuth of a line can be found from its forward azimuth as follows:
If the azimuth of the line is less than 180 degrees then add 180 to the forward azimuth; otherwise
subtract 180 degrees to obtain the back azimuth.
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Bearing of a line (Quadrantal bearing)
Bearing of a line is an acute angle measured from the reference meridian (north or south) in
clock wise or counter clock wise direction towards E or W directions. Bearing angle is never
greater than 900
.
Bearing of A=N 300
E
Bearing of B=S 400
E
Bearing of C=S 600
W
Back bearing: the bearing of a line running in reverse direction.
The bearing of a line in the direction of the survey progress is called a forward bearing, while the
bearing of a line in the opposite direction to the direction of progress is back bearing. The back
bearing can be obtained from the forward bearing by simply changing the letters N to S and E to
W or vice versa.
Relations between bearings and Azimuths
The following rules are followed to convert azimuth to reduced/ quadrantal) bearings;
1. If an azimuth is between 00
and 900
, the line is in the N-E quadrant and the bearing equals the
azimuth.
2. If the azimuth is between 900
and 1800
, the line is in the S-E and the bearing =1800
-AZ.
3. If the azimuth is between 1800
and 2700
, the line is in the S-W and the bearing =AZ-1800
4. If the azimuth is between 2700
and 3600
the line is in the N-W and bearing =3600
–AZ.
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Examples:
1. Calculate the bearings of line with azimuth of a) 750
42’ b) 1120
04’ c) 2590
32’ d) 3390
42’
Soln
: a) AZ=750
42’ N-E quadrant Bearing =AZ=N750
42’E
b) AZ=1120
04’ S-E quadrant Bearing=1800
-1120
04’=S670
56’E
Similarly, work out C and d.
2. Find the back azimuths, Bearing, Back bearing of the following lines from the given forward
azimuths.
a. AB=3200
24’ b. CD=2500
50’
c. DE=1560
09’ d. EF=490
20’
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