Short Notes for Theodolite Traverse Survey & Omitted Measurements | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


 ? Theodolite survey ? Traverse surveying and omitted 
measurements
CHAPTER HIGHLIGHTS
THeodoLite Survey
Introduction
Horizontal angles can be measured by compass. But to get 
precise readings and to measure horizontal and vertical angles 
another instrument called theodolite is used. It can be used for 
laying off  horizontal angles, locating points on line, prolong-
ing survey lines, establishing grades, determining diff erence 
in elevation, setting out curves, etc.
Theodolites
 1. Transit theodolite or transit: In transit theodolite 
the line of sight can be reversed by revolving the 
telescope through 180° in the vertical plane and is 
commonly used.
 2. Non-transit theodolite: These are either plain 
theodolites or Y-theodolites in which the telescope 
cannot be transited.
Parts of Transit Theodolite
 1. Telescope: It is mounted on a spindle known as 
horizontal axis or trunnion axis and internal focusing 
type is widely used.
 2. V ertical circle: It is a circular graduated arc attached 
to the trunnion axis of the telescope and operated by 
means of vertical circle clamp and its corresponding 
slow motion or tangent screw. The circle is graduated 
from 0°–360° in clockwise or it is divided into four 
quadrants.
 3. Index frame (T-frame or vernier frame): It is 
T -shaped frame consisting of a vertical leg known as 
clipping arm and a horizontal bar known as vernier 
arm or index arm whose two extremities are fi tted 
with verniers to read vertical circle. Clip screw is 
used for slight adjustment. Altitude bubble is placed 
on the top of the index frame.
 4. The standards (or A-frame): Two resembling letter 
‘A ’ are mounted on the upper plates. The trunnion axis, 
T-frames and the arm of vertical circle clamp are 
attached to the A-frame.
 5. Levelling head: It consists of two parallel triangular 
plates known as plates. Its functions are: 
 (a) To support main part of instrument.
 (b) To attach the theodolite to the tripod.
 (c) To provide a means for leveling the theodolite.
 6. Lower plate (or scale plate): Lower plate carries 
a lower clamp screw and a corresponding tangent 
screw to rotate the plate and fi x at any position. Size 
of theodolite is represented by the size of scale plate.
 7. Upper plate (or vernier plate): This is connected to 
inner axis and carries two verniers and also an upper 
clamp screw and a corresponding tangent screw.
 • On clamping upper and unclamping the lower clamp, 
instrument rotates on its outer axis without any relative 
motion between the plates.
 • Lower clamp is clamped and upper unclamped, instru-
ment rotates on inner axis with relative motion between 
vernier and scale.
Theodolite, Traverse Survey 
and Omitted Measurements
Part III_Unit 12_Chapter 03.indd   1 5/31/2017   5:01:10 PM
Page 2


 ? Theodolite survey ? Traverse surveying and omitted 
measurements
CHAPTER HIGHLIGHTS
THeodoLite Survey
Introduction
Horizontal angles can be measured by compass. But to get 
precise readings and to measure horizontal and vertical angles 
another instrument called theodolite is used. It can be used for 
laying off  horizontal angles, locating points on line, prolong-
ing survey lines, establishing grades, determining diff erence 
in elevation, setting out curves, etc.
Theodolites
 1. Transit theodolite or transit: In transit theodolite 
the line of sight can be reversed by revolving the 
telescope through 180° in the vertical plane and is 
commonly used.
 2. Non-transit theodolite: These are either plain 
theodolites or Y-theodolites in which the telescope 
cannot be transited.
Parts of Transit Theodolite
 1. Telescope: It is mounted on a spindle known as 
horizontal axis or trunnion axis and internal focusing 
type is widely used.
 2. V ertical circle: It is a circular graduated arc attached 
to the trunnion axis of the telescope and operated by 
means of vertical circle clamp and its corresponding 
slow motion or tangent screw. The circle is graduated 
from 0°–360° in clockwise or it is divided into four 
quadrants.
 3. Index frame (T-frame or vernier frame): It is 
T -shaped frame consisting of a vertical leg known as 
clipping arm and a horizontal bar known as vernier 
arm or index arm whose two extremities are fi tted 
with verniers to read vertical circle. Clip screw is 
used for slight adjustment. Altitude bubble is placed 
on the top of the index frame.
 4. The standards (or A-frame): Two resembling letter 
‘A ’ are mounted on the upper plates. The trunnion axis, 
T-frames and the arm of vertical circle clamp are 
attached to the A-frame.
 5. Levelling head: It consists of two parallel triangular 
plates known as plates. Its functions are: 
 (a) To support main part of instrument.
 (b) To attach the theodolite to the tripod.
 (c) To provide a means for leveling the theodolite.
 6. Lower plate (or scale plate): Lower plate carries 
a lower clamp screw and a corresponding tangent 
screw to rotate the plate and fi x at any position. Size 
of theodolite is represented by the size of scale plate.
 7. Upper plate (or vernier plate): This is connected to 
inner axis and carries two verniers and also an upper 
clamp screw and a corresponding tangent screw.
 • On clamping upper and unclamping the lower clamp, 
instrument rotates on its outer axis without any relative 
motion between the plates.
 • Lower clamp is clamped and upper unclamped, instru-
ment rotates on inner axis with relative motion between 
vernier and scale.
Theodolite, Traverse Survey 
and Omitted Measurements
Part III_Unit 12_Chapter 03.indd   1 5/31/2017   5:01:10 PM
      
 • For using any tangent screw, its corresponding clamp 
screw must be tightened.
 8. Plate levels: These are carried by the upper plates 
which are perpendicular to each other with one of 
them parallel to turnion axis. These help the telescope 
to settle in exact vertical position.
 9. Tripod: It is a stand on which theodolite is mounted
 10. Plumb bob: It is tool a having a cone shaped weight 
attached to a long thread. The weight is hanged using 
thread from centre of tripod stand and centering of 
the odilite is done.
 11. Compass: Simpler theodolites may contain circular 
compass box in the centre of upper plate. It is useful 
when we select north as the reference meridian.
Definitions and Terms
 1. Vertical axis: This is the axis about which the 
instrument can be rotated in a horizontal plane. 
Lower and upper plates rotate about this axis. It is 
also known as azimuth axis.
 2. Horizontal axis or trunnion axis: is the axis about 
which the telescope and the vertical circle rotate in 
the vertical plane.
 3. Line of sight or line of collimation: is the line 
passing through the intersection of the horizontal 
and vertical cross-hairs and the optical centre of the 
object glass and its continuation.
 4. Axis of level tube or bubble line: It is a straight line 
tangential to the longitudinal curve of the level tube at 
its centre. It is horizontal when the bubble is central.
 5. Transiting: also known as plunging or reversing. It is 
the process of turning the telescope in vertical plane 
through 180° about the trunnion axis.
 6. Swinging the telescope: The process of turning the 
telescope in horizontal plane. If rotated clockwise, 
it is called right swing. If rotated in anticlockwise 
direction, it is called left swing.
 7. Face left observation: If the face of vertical circle 
is to the left of the observer, the observation angle is 
called as face left observation.
 8. Face right observation: If the face of vertical circle 
is to the right of the observer.
 9. Telescope normal or direct: Telescope is said to be 
normal when the face of the vertical circle is to the 
left side and the bubble is up.
 10. Telescope inverted or reversed: When face of 
vertical circle is to the right and the bubble down.
 11. Changing face: It is an operation of bringing the face 
of the telescope from left to right and vice versa.
Adjustments in Theodolite
Temporary Adjustments of Theodolite
 • Setting: The instrument to be set over the station, center-
ing by a plumb bob and approximate levelling with the 
help of tripod legs.
 • Levelling: Done using leveling screws or foot screws to 
make the vertical axis truly vertical.
 • Elimination of parallax: Parallax is a condition aris-
ing when the image formed by the objective is not in the 
plane of the cross hairs.
It is done in two steps:
 1. Focusing the eye-piece: For distinct vision of the 
cross hairs.
 2. Focusing the objective: Focusing screw is turned till 
the image appears clear and sharp and is in the plane 
of cross hairs.
Permanent Adjustments of Theodolite
 1. Adjustment of plate level: To make the axis of plate 
bubble perpendicular to the vertical axis when the 
bubble is central.
 2. Adjustment of line of sight: Line of sight should 
coincide with optical axis of the telescope.
 3. Adjustment of horizontal axis: Horizontal axis 
should be perpendicular to the vertical axis. Spire test 
is done.
 4. Adjustment of altitude level and vertical index 
frame: Clip and tangent screws are used for adjusting 
vertical frame and levelling of altitude.
Operations done with Theodolite
Measurement of Horizontal Angle
Repetition Method Used to measure horizontal angle to a 
finer degree of accuracy than that obtained with the least 
count of vernier.
Errors eliminated:
 1. Errors due to eccentricity of verniers and centres are 
eliminated by taking both vernier readings.
 2. Errors due to in adjustment of line of collimation and 
the trunnion axis are eliminated by taking both face 
readings.
 3. Error due to inaccurate graduations are eliminated by 
taking the readings at different parts of the circle.
 4. Errors due to inaccurate bisection of the object, 
eccentric centering, etc., may be to some extent 
counter-balanced in different observations.
	 5. Errors due to slip, displacement of station signals and 
want of verticality of the vertical axis, etc., are not 
eliminated since they are all cumulative.
Direction Method or Reiteration Method This is also 
known as method of series and is suitable for the measure-
ment of the angles of a group having a common vertex point 
and finally the horizon is closed such that sum of angles 
equal to 360°.
 1. Measurement of vertical angles.
 2. Measuring magnetic bearing of a line.
Part III_Unit 12_Chapter 03.indd   2 5/31/2017   5:01:11 PM
Page 3


 ? Theodolite survey ? Traverse surveying and omitted 
measurements
CHAPTER HIGHLIGHTS
THeodoLite Survey
Introduction
Horizontal angles can be measured by compass. But to get 
precise readings and to measure horizontal and vertical angles 
another instrument called theodolite is used. It can be used for 
laying off  horizontal angles, locating points on line, prolong-
ing survey lines, establishing grades, determining diff erence 
in elevation, setting out curves, etc.
Theodolites
 1. Transit theodolite or transit: In transit theodolite 
the line of sight can be reversed by revolving the 
telescope through 180° in the vertical plane and is 
commonly used.
 2. Non-transit theodolite: These are either plain 
theodolites or Y-theodolites in which the telescope 
cannot be transited.
Parts of Transit Theodolite
 1. Telescope: It is mounted on a spindle known as 
horizontal axis or trunnion axis and internal focusing 
type is widely used.
 2. V ertical circle: It is a circular graduated arc attached 
to the trunnion axis of the telescope and operated by 
means of vertical circle clamp and its corresponding 
slow motion or tangent screw. The circle is graduated 
from 0°–360° in clockwise or it is divided into four 
quadrants.
 3. Index frame (T-frame or vernier frame): It is 
T -shaped frame consisting of a vertical leg known as 
clipping arm and a horizontal bar known as vernier 
arm or index arm whose two extremities are fi tted 
with verniers to read vertical circle. Clip screw is 
used for slight adjustment. Altitude bubble is placed 
on the top of the index frame.
 4. The standards (or A-frame): Two resembling letter 
‘A ’ are mounted on the upper plates. The trunnion axis, 
T-frames and the arm of vertical circle clamp are 
attached to the A-frame.
 5. Levelling head: It consists of two parallel triangular 
plates known as plates. Its functions are: 
 (a) To support main part of instrument.
 (b) To attach the theodolite to the tripod.
 (c) To provide a means for leveling the theodolite.
 6. Lower plate (or scale plate): Lower plate carries 
a lower clamp screw and a corresponding tangent 
screw to rotate the plate and fi x at any position. Size 
of theodolite is represented by the size of scale plate.
 7. Upper plate (or vernier plate): This is connected to 
inner axis and carries two verniers and also an upper 
clamp screw and a corresponding tangent screw.
 • On clamping upper and unclamping the lower clamp, 
instrument rotates on its outer axis without any relative 
motion between the plates.
 • Lower clamp is clamped and upper unclamped, instru-
ment rotates on inner axis with relative motion between 
vernier and scale.
Theodolite, Traverse Survey 
and Omitted Measurements
Part III_Unit 12_Chapter 03.indd   1 5/31/2017   5:01:10 PM
      
 • For using any tangent screw, its corresponding clamp 
screw must be tightened.
 8. Plate levels: These are carried by the upper plates 
which are perpendicular to each other with one of 
them parallel to turnion axis. These help the telescope 
to settle in exact vertical position.
 9. Tripod: It is a stand on which theodolite is mounted
 10. Plumb bob: It is tool a having a cone shaped weight 
attached to a long thread. The weight is hanged using 
thread from centre of tripod stand and centering of 
the odilite is done.
 11. Compass: Simpler theodolites may contain circular 
compass box in the centre of upper plate. It is useful 
when we select north as the reference meridian.
Definitions and Terms
 1. Vertical axis: This is the axis about which the 
instrument can be rotated in a horizontal plane. 
Lower and upper plates rotate about this axis. It is 
also known as azimuth axis.
 2. Horizontal axis or trunnion axis: is the axis about 
which the telescope and the vertical circle rotate in 
the vertical plane.
 3. Line of sight or line of collimation: is the line 
passing through the intersection of the horizontal 
and vertical cross-hairs and the optical centre of the 
object glass and its continuation.
 4. Axis of level tube or bubble line: It is a straight line 
tangential to the longitudinal curve of the level tube at 
its centre. It is horizontal when the bubble is central.
 5. Transiting: also known as plunging or reversing. It is 
the process of turning the telescope in vertical plane 
through 180° about the trunnion axis.
 6. Swinging the telescope: The process of turning the 
telescope in horizontal plane. If rotated clockwise, 
it is called right swing. If rotated in anticlockwise 
direction, it is called left swing.
 7. Face left observation: If the face of vertical circle 
is to the left of the observer, the observation angle is 
called as face left observation.
 8. Face right observation: If the face of vertical circle 
is to the right of the observer.
 9. Telescope normal or direct: Telescope is said to be 
normal when the face of the vertical circle is to the 
left side and the bubble is up.
 10. Telescope inverted or reversed: When face of 
vertical circle is to the right and the bubble down.
 11. Changing face: It is an operation of bringing the face 
of the telescope from left to right and vice versa.
Adjustments in Theodolite
Temporary Adjustments of Theodolite
 • Setting: The instrument to be set over the station, center-
ing by a plumb bob and approximate levelling with the 
help of tripod legs.
 • Levelling: Done using leveling screws or foot screws to 
make the vertical axis truly vertical.
 • Elimination of parallax: Parallax is a condition aris-
ing when the image formed by the objective is not in the 
plane of the cross hairs.
It is done in two steps:
 1. Focusing the eye-piece: For distinct vision of the 
cross hairs.
 2. Focusing the objective: Focusing screw is turned till 
the image appears clear and sharp and is in the plane 
of cross hairs.
Permanent Adjustments of Theodolite
 1. Adjustment of plate level: To make the axis of plate 
bubble perpendicular to the vertical axis when the 
bubble is central.
 2. Adjustment of line of sight: Line of sight should 
coincide with optical axis of the telescope.
 3. Adjustment of horizontal axis: Horizontal axis 
should be perpendicular to the vertical axis. Spire test 
is done.
 4. Adjustment of altitude level and vertical index 
frame: Clip and tangent screws are used for adjusting 
vertical frame and levelling of altitude.
Operations done with Theodolite
Measurement of Horizontal Angle
Repetition Method Used to measure horizontal angle to a 
finer degree of accuracy than that obtained with the least 
count of vernier.
Errors eliminated:
 1. Errors due to eccentricity of verniers and centres are 
eliminated by taking both vernier readings.
 2. Errors due to in adjustment of line of collimation and 
the trunnion axis are eliminated by taking both face 
readings.
 3. Error due to inaccurate graduations are eliminated by 
taking the readings at different parts of the circle.
 4. Errors due to inaccurate bisection of the object, 
eccentric centering, etc., may be to some extent 
counter-balanced in different observations.
	 5. Errors due to slip, displacement of station signals and 
want of verticality of the vertical axis, etc., are not 
eliminated since they are all cumulative.
Direction Method or Reiteration Method This is also 
known as method of series and is suitable for the measure-
ment of the angles of a group having a common vertex point 
and finally the horizon is closed such that sum of angles 
equal to 360°.
 1. Measurement of vertical angles.
 2. Measuring magnetic bearing of a line.
Part III_Unit 12_Chapter 03.indd   2 5/31/2017   5:01:11 PM
    
 3. Measuring direct angles and deflection angles.
 4. To prolong a straight line.
 5. To locate point of intersection of two straight lines.
 6. To lay off a horizontal angle.
 7. To lay off an angle by repetition.
Fundamental Lines and Their 
Desired Relations
 1. Vertical axis
 2. Horizontal axis (trunnion or transit axis)
 3. Line of collimation (line of sight)
 4. Axis of plate level
 5. Axis of altitude level
When the theodolite is in proper adjustment:
 • The axis of the plate level must lie in a plane perpendicu-
lar to the vertical axis.
 • The line of collimation must be perpendicular to the hori-
zontal axis at its intersection with the vertical axis.
 • Horizontal axis must be perpendicular to vertical axis.
 • Axis of altitude level (telescope level) must be parallel to 
line of collimation.
 • Vertical circle vernier must read zero when the line of 
collimation is horizontal.
Sources of Error in Theodolite Work
 1. Instrumental
 2. Personal
 3. Natural
Instrumental Errors
These are due to: 
 1. Imperfect adjustment of the instrument.
 2. Structural defects in instrument
 3. Imperfections due to wear.
 • Error due to imperfect adjustment of plate levels: 
The error can be eliminated only by careful leveling 
with respect to the altitude bubble if it is in adjust-
ment and can be eliminated by double sighting.
 • Error due to line of collimation not being perpen-
dicular to the horizontal axis:
  Error, e = b sec a 	 	b = Error in collimation
  a = Inclination
This error can be eliminated by taking both face 
observations.
 • Error due to horizontal axis not being perpendicular to 
the vertical axis:
Error, e = b tan a This error can be eliminated by taking both face 
observations.
 • Error due to non-parallelism of the axis of telescope level 
and the line of collimation can be eliminated by taking 
both face observations.
 • Error due to eccentricity of inner and outer axes can 
be eliminated by taking the mean of the two vernier 
readings.
 • Error due to imperfect graduations can be eliminated by 
taking mean of several readings distributed over different 
portions of the graduated circle.
 • Error due to eccentricity of verniers can be eliminated 
by reading both the verniers and taking the mean of the 
two.
Personal Errors
 1. Errors in manipulation:
 (a) Inaccurate centering
 (b) Inaccurate levelling
 (c) Slip
 (d) Manipulating wrong tangent screw.
 2. Errors in sighting and reading:
 (a) Inaccurate bisection of points observed.
 (b) Mistakes in setting the vernier.
 (c) Parallax: Due to parallax, accurate bisection is 
not possible. Error can be eliminated by focusing 
eye-piece and objective.
Natural Errors
 1. Unequal atmospheric refraction due to high 
temperature.
 2. Unequal expansion of parts of telescope and circles 
due to temperature change.
 3. Unequal settlement of tripod.
 4. Wind producing vibrations.
Double Sighting
Measurement of horizontal angle or vertical angle twice; 
once with the telescope in normal condition and once 
with the telescope in inverted condition is called double 
sighting.
Traverse Surveying and  
Omitted Measurements
 1. Introduction: Traversing is that type of survey in 
which a number of connected survey lines form the 
framework and the directions and the lengths of the 
survey lines are measured with the help of an angle 
measuring instrument (compass or theodolite) and a 
tape (or chain) respectively.
 • When the lines form a circuit which ends at the 
starting point, it is known as closed traverse. 
Part III_Unit 12_Chapter 03.indd   3 5/31/2017   5:01:11 PM
Page 4


 ? Theodolite survey ? Traverse surveying and omitted 
measurements
CHAPTER HIGHLIGHTS
THeodoLite Survey
Introduction
Horizontal angles can be measured by compass. But to get 
precise readings and to measure horizontal and vertical angles 
another instrument called theodolite is used. It can be used for 
laying off  horizontal angles, locating points on line, prolong-
ing survey lines, establishing grades, determining diff erence 
in elevation, setting out curves, etc.
Theodolites
 1. Transit theodolite or transit: In transit theodolite 
the line of sight can be reversed by revolving the 
telescope through 180° in the vertical plane and is 
commonly used.
 2. Non-transit theodolite: These are either plain 
theodolites or Y-theodolites in which the telescope 
cannot be transited.
Parts of Transit Theodolite
 1. Telescope: It is mounted on a spindle known as 
horizontal axis or trunnion axis and internal focusing 
type is widely used.
 2. V ertical circle: It is a circular graduated arc attached 
to the trunnion axis of the telescope and operated by 
means of vertical circle clamp and its corresponding 
slow motion or tangent screw. The circle is graduated 
from 0°–360° in clockwise or it is divided into four 
quadrants.
 3. Index frame (T-frame or vernier frame): It is 
T -shaped frame consisting of a vertical leg known as 
clipping arm and a horizontal bar known as vernier 
arm or index arm whose two extremities are fi tted 
with verniers to read vertical circle. Clip screw is 
used for slight adjustment. Altitude bubble is placed 
on the top of the index frame.
 4. The standards (or A-frame): Two resembling letter 
‘A ’ are mounted on the upper plates. The trunnion axis, 
T-frames and the arm of vertical circle clamp are 
attached to the A-frame.
 5. Levelling head: It consists of two parallel triangular 
plates known as plates. Its functions are: 
 (a) To support main part of instrument.
 (b) To attach the theodolite to the tripod.
 (c) To provide a means for leveling the theodolite.
 6. Lower plate (or scale plate): Lower plate carries 
a lower clamp screw and a corresponding tangent 
screw to rotate the plate and fi x at any position. Size 
of theodolite is represented by the size of scale plate.
 7. Upper plate (or vernier plate): This is connected to 
inner axis and carries two verniers and also an upper 
clamp screw and a corresponding tangent screw.
 • On clamping upper and unclamping the lower clamp, 
instrument rotates on its outer axis without any relative 
motion between the plates.
 • Lower clamp is clamped and upper unclamped, instru-
ment rotates on inner axis with relative motion between 
vernier and scale.
Theodolite, Traverse Survey 
and Omitted Measurements
Part III_Unit 12_Chapter 03.indd   1 5/31/2017   5:01:10 PM
      
 • For using any tangent screw, its corresponding clamp 
screw must be tightened.
 8. Plate levels: These are carried by the upper plates 
which are perpendicular to each other with one of 
them parallel to turnion axis. These help the telescope 
to settle in exact vertical position.
 9. Tripod: It is a stand on which theodolite is mounted
 10. Plumb bob: It is tool a having a cone shaped weight 
attached to a long thread. The weight is hanged using 
thread from centre of tripod stand and centering of 
the odilite is done.
 11. Compass: Simpler theodolites may contain circular 
compass box in the centre of upper plate. It is useful 
when we select north as the reference meridian.
Definitions and Terms
 1. Vertical axis: This is the axis about which the 
instrument can be rotated in a horizontal plane. 
Lower and upper plates rotate about this axis. It is 
also known as azimuth axis.
 2. Horizontal axis or trunnion axis: is the axis about 
which the telescope and the vertical circle rotate in 
the vertical plane.
 3. Line of sight or line of collimation: is the line 
passing through the intersection of the horizontal 
and vertical cross-hairs and the optical centre of the 
object glass and its continuation.
 4. Axis of level tube or bubble line: It is a straight line 
tangential to the longitudinal curve of the level tube at 
its centre. It is horizontal when the bubble is central.
 5. Transiting: also known as plunging or reversing. It is 
the process of turning the telescope in vertical plane 
through 180° about the trunnion axis.
 6. Swinging the telescope: The process of turning the 
telescope in horizontal plane. If rotated clockwise, 
it is called right swing. If rotated in anticlockwise 
direction, it is called left swing.
 7. Face left observation: If the face of vertical circle 
is to the left of the observer, the observation angle is 
called as face left observation.
 8. Face right observation: If the face of vertical circle 
is to the right of the observer.
 9. Telescope normal or direct: Telescope is said to be 
normal when the face of the vertical circle is to the 
left side and the bubble is up.
 10. Telescope inverted or reversed: When face of 
vertical circle is to the right and the bubble down.
 11. Changing face: It is an operation of bringing the face 
of the telescope from left to right and vice versa.
Adjustments in Theodolite
Temporary Adjustments of Theodolite
 • Setting: The instrument to be set over the station, center-
ing by a plumb bob and approximate levelling with the 
help of tripod legs.
 • Levelling: Done using leveling screws or foot screws to 
make the vertical axis truly vertical.
 • Elimination of parallax: Parallax is a condition aris-
ing when the image formed by the objective is not in the 
plane of the cross hairs.
It is done in two steps:
 1. Focusing the eye-piece: For distinct vision of the 
cross hairs.
 2. Focusing the objective: Focusing screw is turned till 
the image appears clear and sharp and is in the plane 
of cross hairs.
Permanent Adjustments of Theodolite
 1. Adjustment of plate level: To make the axis of plate 
bubble perpendicular to the vertical axis when the 
bubble is central.
 2. Adjustment of line of sight: Line of sight should 
coincide with optical axis of the telescope.
 3. Adjustment of horizontal axis: Horizontal axis 
should be perpendicular to the vertical axis. Spire test 
is done.
 4. Adjustment of altitude level and vertical index 
frame: Clip and tangent screws are used for adjusting 
vertical frame and levelling of altitude.
Operations done with Theodolite
Measurement of Horizontal Angle
Repetition Method Used to measure horizontal angle to a 
finer degree of accuracy than that obtained with the least 
count of vernier.
Errors eliminated:
 1. Errors due to eccentricity of verniers and centres are 
eliminated by taking both vernier readings.
 2. Errors due to in adjustment of line of collimation and 
the trunnion axis are eliminated by taking both face 
readings.
 3. Error due to inaccurate graduations are eliminated by 
taking the readings at different parts of the circle.
 4. Errors due to inaccurate bisection of the object, 
eccentric centering, etc., may be to some extent 
counter-balanced in different observations.
	 5. Errors due to slip, displacement of station signals and 
want of verticality of the vertical axis, etc., are not 
eliminated since they are all cumulative.
Direction Method or Reiteration Method This is also 
known as method of series and is suitable for the measure-
ment of the angles of a group having a common vertex point 
and finally the horizon is closed such that sum of angles 
equal to 360°.
 1. Measurement of vertical angles.
 2. Measuring magnetic bearing of a line.
Part III_Unit 12_Chapter 03.indd   2 5/31/2017   5:01:11 PM
    
 3. Measuring direct angles and deflection angles.
 4. To prolong a straight line.
 5. To locate point of intersection of two straight lines.
 6. To lay off a horizontal angle.
 7. To lay off an angle by repetition.
Fundamental Lines and Their 
Desired Relations
 1. Vertical axis
 2. Horizontal axis (trunnion or transit axis)
 3. Line of collimation (line of sight)
 4. Axis of plate level
 5. Axis of altitude level
When the theodolite is in proper adjustment:
 • The axis of the plate level must lie in a plane perpendicu-
lar to the vertical axis.
 • The line of collimation must be perpendicular to the hori-
zontal axis at its intersection with the vertical axis.
 • Horizontal axis must be perpendicular to vertical axis.
 • Axis of altitude level (telescope level) must be parallel to 
line of collimation.
 • Vertical circle vernier must read zero when the line of 
collimation is horizontal.
Sources of Error in Theodolite Work
 1. Instrumental
 2. Personal
 3. Natural
Instrumental Errors
These are due to: 
 1. Imperfect adjustment of the instrument.
 2. Structural defects in instrument
 3. Imperfections due to wear.
 • Error due to imperfect adjustment of plate levels: 
The error can be eliminated only by careful leveling 
with respect to the altitude bubble if it is in adjust-
ment and can be eliminated by double sighting.
 • Error due to line of collimation not being perpen-
dicular to the horizontal axis:
  Error, e = b sec a 	 	b = Error in collimation
  a = Inclination
This error can be eliminated by taking both face 
observations.
 • Error due to horizontal axis not being perpendicular to 
the vertical axis:
Error, e = b tan a This error can be eliminated by taking both face 
observations.
 • Error due to non-parallelism of the axis of telescope level 
and the line of collimation can be eliminated by taking 
both face observations.
 • Error due to eccentricity of inner and outer axes can 
be eliminated by taking the mean of the two vernier 
readings.
 • Error due to imperfect graduations can be eliminated by 
taking mean of several readings distributed over different 
portions of the graduated circle.
 • Error due to eccentricity of verniers can be eliminated 
by reading both the verniers and taking the mean of the 
two.
Personal Errors
 1. Errors in manipulation:
 (a) Inaccurate centering
 (b) Inaccurate levelling
 (c) Slip
 (d) Manipulating wrong tangent screw.
 2. Errors in sighting and reading:
 (a) Inaccurate bisection of points observed.
 (b) Mistakes in setting the vernier.
 (c) Parallax: Due to parallax, accurate bisection is 
not possible. Error can be eliminated by focusing 
eye-piece and objective.
Natural Errors
 1. Unequal atmospheric refraction due to high 
temperature.
 2. Unequal expansion of parts of telescope and circles 
due to temperature change.
 3. Unequal settlement of tripod.
 4. Wind producing vibrations.
Double Sighting
Measurement of horizontal angle or vertical angle twice; 
once with the telescope in normal condition and once 
with the telescope in inverted condition is called double 
sighting.
Traverse Surveying and  
Omitted Measurements
 1. Introduction: Traversing is that type of survey in 
which a number of connected survey lines form the 
framework and the directions and the lengths of the 
survey lines are measured with the help of an angle 
measuring instrument (compass or theodolite) and a 
tape (or chain) respectively.
 • When the lines form a circuit which ends at the 
starting point, it is known as closed traverse. 
Part III_Unit 12_Chapter 03.indd   3 5/31/2017   5:01:11 PM
      
This is suitable for locating the boundaries of 
lakes, woods etc., and for the survey of large areas.
 • If the lines of survey ends at a point other than the 
starting point it is said be an open traverse. This is 
suitable for surveying a long narrow strip of land 
as required for a road or canal or the coastline.
Methods of Traversing
 1. Chain traversing
 2. Chain and compass traversing (loose needle method)
 3. Transit type traversing
 (a) By fast needle method
 (b) By measurement of angles between the lines
 4. Plane table traversing
Traverse surveying is not limited to any particular geometri-
cal figure as in the case of chain surveying which has sys-
tem of connected triangles.
Chain Traversing
 • In this, traversing is done with the chain and tape. 
Directions of the lines are fixed entirely by linear or tie 
measurements known as chain angles.
 • Not suitable for accurate work and generally chain tra-
versing is not used if an angle measurement instrument 
such as compass or theodolite are available.
Chain and Compass Traversing
Linear measurements are done with the help of chain or 
tape and the bearings are measured with compass. Both FB 
and BB are observed at each station.
Transit Type Traversing
 1. Fast needle method:
 • Magnetic bearings of traverse lines are measured 
by a theodolite fitted with a compass with refer -
ence to the direction of magnetic meridian estab-
lished at the first station.
 • In direct method with transiting, telescope will be 
normal at one station and inverted at the next station.
 • Direct method without transiting is the most satis-
factory method in which 180° correction is neces-
sary only at 2nd, 4th and 6th station and so on. Add 
180° if the reading of vernier < 180°. Subtract 180° 
if it is more than 180°.
 2. Direct observation of angles: Angles are directly 
measured with theodolites. Traversing is done by:
 (a) Included angles
 (b) Deflection angles
 • Deflection angle is the angle, which a survey 
line makes with the prolongation of the pre-
vious line. More suitable for survey of roads, 
railways, pipe-lines, etc.
 • If the direction of progress is counter- 
clockwise and the angles measured clockwise 
are directly the interior angles. 
 • If the direction of progress is clockwise and 
hence the angles measured clockwise are exte-
rior angles.
Checks in Closed Traverse
Two kinds of errors are involved in traversing.
Linear Error
This shall be rectified by measuring each survey line for 
second time. Preferably in the reverse direction on different 
dates and by different parties.
Angular Error
 1. Traverse by included angles:
 • If the interior angles are measured, sum of interior 
angles = (2N – 4) 90°.
 • If the exterior angles are measured, sum of exterior 
angles = (2N + 4) 90°.
N = Number of sides of the traverse.
 2. Traverse by deflection angles:
S Deflection Angles = 360° 
 [Right hand deflection angles as ‘ +’, Left hand deflec-
tion angles as ‘–’]
 3. Traverse by direct observation of bearings: Last line 
FB = (Last line BB measured at the initial station ± 180°).
Plotting a Traverse Survey
Angle and Distance Method
This method is suitable for small surveys. Distance between 
stations are laid off to scale and angles are plotted by one of 
the following methods:
 1. By protractor
 2. By the tangent of the angle
 3. By the chord of the angle
Coordinate Method
This method is the most practical and accurate one for plot-
ting traverse or any other extensive system of horizontal 
control. Survey stations are plotted by calculating their co-
ordinates. Advantage of this method is that the closing error 
can be eliminated by balancing prior to plotting.
 1. Dependent or consecutive co-ordinates (latitude 
or departure):
 • Latitude (L) of a survey line may be defined as its 
co-ordinate length measured parallel to an assumed 
meridian direction.
 • Latitude is positive when measured northward 
(upward) and is termed as northing. It is nega-
tive when measured southward (downward) and is 
termed as southing.
Part III_Unit 12_Chapter 03.indd   4 5/31/2017   5:01:11 PM
Page 5


 ? Theodolite survey ? Traverse surveying and omitted 
measurements
CHAPTER HIGHLIGHTS
THeodoLite Survey
Introduction
Horizontal angles can be measured by compass. But to get 
precise readings and to measure horizontal and vertical angles 
another instrument called theodolite is used. It can be used for 
laying off  horizontal angles, locating points on line, prolong-
ing survey lines, establishing grades, determining diff erence 
in elevation, setting out curves, etc.
Theodolites
 1. Transit theodolite or transit: In transit theodolite 
the line of sight can be reversed by revolving the 
telescope through 180° in the vertical plane and is 
commonly used.
 2. Non-transit theodolite: These are either plain 
theodolites or Y-theodolites in which the telescope 
cannot be transited.
Parts of Transit Theodolite
 1. Telescope: It is mounted on a spindle known as 
horizontal axis or trunnion axis and internal focusing 
type is widely used.
 2. V ertical circle: It is a circular graduated arc attached 
to the trunnion axis of the telescope and operated by 
means of vertical circle clamp and its corresponding 
slow motion or tangent screw. The circle is graduated 
from 0°–360° in clockwise or it is divided into four 
quadrants.
 3. Index frame (T-frame or vernier frame): It is 
T -shaped frame consisting of a vertical leg known as 
clipping arm and a horizontal bar known as vernier 
arm or index arm whose two extremities are fi tted 
with verniers to read vertical circle. Clip screw is 
used for slight adjustment. Altitude bubble is placed 
on the top of the index frame.
 4. The standards (or A-frame): Two resembling letter 
‘A ’ are mounted on the upper plates. The trunnion axis, 
T-frames and the arm of vertical circle clamp are 
attached to the A-frame.
 5. Levelling head: It consists of two parallel triangular 
plates known as plates. Its functions are: 
 (a) To support main part of instrument.
 (b) To attach the theodolite to the tripod.
 (c) To provide a means for leveling the theodolite.
 6. Lower plate (or scale plate): Lower plate carries 
a lower clamp screw and a corresponding tangent 
screw to rotate the plate and fi x at any position. Size 
of theodolite is represented by the size of scale plate.
 7. Upper plate (or vernier plate): This is connected to 
inner axis and carries two verniers and also an upper 
clamp screw and a corresponding tangent screw.
 • On clamping upper and unclamping the lower clamp, 
instrument rotates on its outer axis without any relative 
motion between the plates.
 • Lower clamp is clamped and upper unclamped, instru-
ment rotates on inner axis with relative motion between 
vernier and scale.
Theodolite, Traverse Survey 
and Omitted Measurements
Part III_Unit 12_Chapter 03.indd   1 5/31/2017   5:01:10 PM
      
 • For using any tangent screw, its corresponding clamp 
screw must be tightened.
 8. Plate levels: These are carried by the upper plates 
which are perpendicular to each other with one of 
them parallel to turnion axis. These help the telescope 
to settle in exact vertical position.
 9. Tripod: It is a stand on which theodolite is mounted
 10. Plumb bob: It is tool a having a cone shaped weight 
attached to a long thread. The weight is hanged using 
thread from centre of tripod stand and centering of 
the odilite is done.
 11. Compass: Simpler theodolites may contain circular 
compass box in the centre of upper plate. It is useful 
when we select north as the reference meridian.
Definitions and Terms
 1. Vertical axis: This is the axis about which the 
instrument can be rotated in a horizontal plane. 
Lower and upper plates rotate about this axis. It is 
also known as azimuth axis.
 2. Horizontal axis or trunnion axis: is the axis about 
which the telescope and the vertical circle rotate in 
the vertical plane.
 3. Line of sight or line of collimation: is the line 
passing through the intersection of the horizontal 
and vertical cross-hairs and the optical centre of the 
object glass and its continuation.
 4. Axis of level tube or bubble line: It is a straight line 
tangential to the longitudinal curve of the level tube at 
its centre. It is horizontal when the bubble is central.
 5. Transiting: also known as plunging or reversing. It is 
the process of turning the telescope in vertical plane 
through 180° about the trunnion axis.
 6. Swinging the telescope: The process of turning the 
telescope in horizontal plane. If rotated clockwise, 
it is called right swing. If rotated in anticlockwise 
direction, it is called left swing.
 7. Face left observation: If the face of vertical circle 
is to the left of the observer, the observation angle is 
called as face left observation.
 8. Face right observation: If the face of vertical circle 
is to the right of the observer.
 9. Telescope normal or direct: Telescope is said to be 
normal when the face of the vertical circle is to the 
left side and the bubble is up.
 10. Telescope inverted or reversed: When face of 
vertical circle is to the right and the bubble down.
 11. Changing face: It is an operation of bringing the face 
of the telescope from left to right and vice versa.
Adjustments in Theodolite
Temporary Adjustments of Theodolite
 • Setting: The instrument to be set over the station, center-
ing by a plumb bob and approximate levelling with the 
help of tripod legs.
 • Levelling: Done using leveling screws or foot screws to 
make the vertical axis truly vertical.
 • Elimination of parallax: Parallax is a condition aris-
ing when the image formed by the objective is not in the 
plane of the cross hairs.
It is done in two steps:
 1. Focusing the eye-piece: For distinct vision of the 
cross hairs.
 2. Focusing the objective: Focusing screw is turned till 
the image appears clear and sharp and is in the plane 
of cross hairs.
Permanent Adjustments of Theodolite
 1. Adjustment of plate level: To make the axis of plate 
bubble perpendicular to the vertical axis when the 
bubble is central.
 2. Adjustment of line of sight: Line of sight should 
coincide with optical axis of the telescope.
 3. Adjustment of horizontal axis: Horizontal axis 
should be perpendicular to the vertical axis. Spire test 
is done.
 4. Adjustment of altitude level and vertical index 
frame: Clip and tangent screws are used for adjusting 
vertical frame and levelling of altitude.
Operations done with Theodolite
Measurement of Horizontal Angle
Repetition Method Used to measure horizontal angle to a 
finer degree of accuracy than that obtained with the least 
count of vernier.
Errors eliminated:
 1. Errors due to eccentricity of verniers and centres are 
eliminated by taking both vernier readings.
 2. Errors due to in adjustment of line of collimation and 
the trunnion axis are eliminated by taking both face 
readings.
 3. Error due to inaccurate graduations are eliminated by 
taking the readings at different parts of the circle.
 4. Errors due to inaccurate bisection of the object, 
eccentric centering, etc., may be to some extent 
counter-balanced in different observations.
	 5. Errors due to slip, displacement of station signals and 
want of verticality of the vertical axis, etc., are not 
eliminated since they are all cumulative.
Direction Method or Reiteration Method This is also 
known as method of series and is suitable for the measure-
ment of the angles of a group having a common vertex point 
and finally the horizon is closed such that sum of angles 
equal to 360°.
 1. Measurement of vertical angles.
 2. Measuring magnetic bearing of a line.
Part III_Unit 12_Chapter 03.indd   2 5/31/2017   5:01:11 PM
    
 3. Measuring direct angles and deflection angles.
 4. To prolong a straight line.
 5. To locate point of intersection of two straight lines.
 6. To lay off a horizontal angle.
 7. To lay off an angle by repetition.
Fundamental Lines and Their 
Desired Relations
 1. Vertical axis
 2. Horizontal axis (trunnion or transit axis)
 3. Line of collimation (line of sight)
 4. Axis of plate level
 5. Axis of altitude level
When the theodolite is in proper adjustment:
 • The axis of the plate level must lie in a plane perpendicu-
lar to the vertical axis.
 • The line of collimation must be perpendicular to the hori-
zontal axis at its intersection with the vertical axis.
 • Horizontal axis must be perpendicular to vertical axis.
 • Axis of altitude level (telescope level) must be parallel to 
line of collimation.
 • Vertical circle vernier must read zero when the line of 
collimation is horizontal.
Sources of Error in Theodolite Work
 1. Instrumental
 2. Personal
 3. Natural
Instrumental Errors
These are due to: 
 1. Imperfect adjustment of the instrument.
 2. Structural defects in instrument
 3. Imperfections due to wear.
 • Error due to imperfect adjustment of plate levels: 
The error can be eliminated only by careful leveling 
with respect to the altitude bubble if it is in adjust-
ment and can be eliminated by double sighting.
 • Error due to line of collimation not being perpen-
dicular to the horizontal axis:
  Error, e = b sec a 	 	b = Error in collimation
  a = Inclination
This error can be eliminated by taking both face 
observations.
 • Error due to horizontal axis not being perpendicular to 
the vertical axis:
Error, e = b tan a This error can be eliminated by taking both face 
observations.
 • Error due to non-parallelism of the axis of telescope level 
and the line of collimation can be eliminated by taking 
both face observations.
 • Error due to eccentricity of inner and outer axes can 
be eliminated by taking the mean of the two vernier 
readings.
 • Error due to imperfect graduations can be eliminated by 
taking mean of several readings distributed over different 
portions of the graduated circle.
 • Error due to eccentricity of verniers can be eliminated 
by reading both the verniers and taking the mean of the 
two.
Personal Errors
 1. Errors in manipulation:
 (a) Inaccurate centering
 (b) Inaccurate levelling
 (c) Slip
 (d) Manipulating wrong tangent screw.
 2. Errors in sighting and reading:
 (a) Inaccurate bisection of points observed.
 (b) Mistakes in setting the vernier.
 (c) Parallax: Due to parallax, accurate bisection is 
not possible. Error can be eliminated by focusing 
eye-piece and objective.
Natural Errors
 1. Unequal atmospheric refraction due to high 
temperature.
 2. Unequal expansion of parts of telescope and circles 
due to temperature change.
 3. Unequal settlement of tripod.
 4. Wind producing vibrations.
Double Sighting
Measurement of horizontal angle or vertical angle twice; 
once with the telescope in normal condition and once 
with the telescope in inverted condition is called double 
sighting.
Traverse Surveying and  
Omitted Measurements
 1. Introduction: Traversing is that type of survey in 
which a number of connected survey lines form the 
framework and the directions and the lengths of the 
survey lines are measured with the help of an angle 
measuring instrument (compass or theodolite) and a 
tape (or chain) respectively.
 • When the lines form a circuit which ends at the 
starting point, it is known as closed traverse. 
Part III_Unit 12_Chapter 03.indd   3 5/31/2017   5:01:11 PM
      
This is suitable for locating the boundaries of 
lakes, woods etc., and for the survey of large areas.
 • If the lines of survey ends at a point other than the 
starting point it is said be an open traverse. This is 
suitable for surveying a long narrow strip of land 
as required for a road or canal or the coastline.
Methods of Traversing
 1. Chain traversing
 2. Chain and compass traversing (loose needle method)
 3. Transit type traversing
 (a) By fast needle method
 (b) By measurement of angles between the lines
 4. Plane table traversing
Traverse surveying is not limited to any particular geometri-
cal figure as in the case of chain surveying which has sys-
tem of connected triangles.
Chain Traversing
 • In this, traversing is done with the chain and tape. 
Directions of the lines are fixed entirely by linear or tie 
measurements known as chain angles.
 • Not suitable for accurate work and generally chain tra-
versing is not used if an angle measurement instrument 
such as compass or theodolite are available.
Chain and Compass Traversing
Linear measurements are done with the help of chain or 
tape and the bearings are measured with compass. Both FB 
and BB are observed at each station.
Transit Type Traversing
 1. Fast needle method:
 • Magnetic bearings of traverse lines are measured 
by a theodolite fitted with a compass with refer -
ence to the direction of magnetic meridian estab-
lished at the first station.
 • In direct method with transiting, telescope will be 
normal at one station and inverted at the next station.
 • Direct method without transiting is the most satis-
factory method in which 180° correction is neces-
sary only at 2nd, 4th and 6th station and so on. Add 
180° if the reading of vernier < 180°. Subtract 180° 
if it is more than 180°.
 2. Direct observation of angles: Angles are directly 
measured with theodolites. Traversing is done by:
 (a) Included angles
 (b) Deflection angles
 • Deflection angle is the angle, which a survey 
line makes with the prolongation of the pre-
vious line. More suitable for survey of roads, 
railways, pipe-lines, etc.
 • If the direction of progress is counter- 
clockwise and the angles measured clockwise 
are directly the interior angles. 
 • If the direction of progress is clockwise and 
hence the angles measured clockwise are exte-
rior angles.
Checks in Closed Traverse
Two kinds of errors are involved in traversing.
Linear Error
This shall be rectified by measuring each survey line for 
second time. Preferably in the reverse direction on different 
dates and by different parties.
Angular Error
 1. Traverse by included angles:
 • If the interior angles are measured, sum of interior 
angles = (2N – 4) 90°.
 • If the exterior angles are measured, sum of exterior 
angles = (2N + 4) 90°.
N = Number of sides of the traverse.
 2. Traverse by deflection angles:
S Deflection Angles = 360° 
 [Right hand deflection angles as ‘ +’, Left hand deflec-
tion angles as ‘–’]
 3. Traverse by direct observation of bearings: Last line 
FB = (Last line BB measured at the initial station ± 180°).
Plotting a Traverse Survey
Angle and Distance Method
This method is suitable for small surveys. Distance between 
stations are laid off to scale and angles are plotted by one of 
the following methods:
 1. By protractor
 2. By the tangent of the angle
 3. By the chord of the angle
Coordinate Method
This method is the most practical and accurate one for plot-
ting traverse or any other extensive system of horizontal 
control. Survey stations are plotted by calculating their co-
ordinates. Advantage of this method is that the closing error 
can be eliminated by balancing prior to plotting.
 1. Dependent or consecutive co-ordinates (latitude 
or departure):
 • Latitude (L) of a survey line may be defined as its 
co-ordinate length measured parallel to an assumed 
meridian direction.
 • Latitude is positive when measured northward 
(upward) and is termed as northing. It is nega-
tive when measured southward (downward) and is 
termed as southing.
Part III_Unit 12_Chapter 03.indd   4 5/31/2017   5:01:11 PM
    
 • Departure (D) of survey line may be defined as its 
co-ordinate length measured at right angles to the 
meridian direction.
 • Departure of the line is positive when measured 
eastward and is termed as easting. It is negative 
when measured westward and is termed as westing.
From the figure,
L = + lcos q D = + lsin q I
D = l sin?
(+, -) (+, +)
(-, +) (-, -)
L = l cos?
?
 2. Independent co-ordinates: Total latitude and 
departure of any point with respect to a common 
origin are known as independent co-ordinates or total 
co-ordinates of the point.
Closing Error
If a closed traverse is plotted according to the field measure-
ments, the end point of the traverse will not coincide exactly with 
the starting point owing to the errors in the field measurements 
of angles and distances. This error is known as closing error.
Closing error,
eL D =+ () () SS
22
Direction of closing error, tand =
?
?
D
L
Relative error of closure
= 
Error of closure 
Perimeter of traverse
= 
e
p
p
e
=
1
d
Closing
error
SL
SD
 • For a closed traverse, SL = SD = 0
SOLVED EXAMPLES
Example 1
A closed traverse was conducted round an obstacle and the 
following observations were made and length and bearing 
of DA have been omitted. Calculate length and bearing of 
the line DA.
Line Length (m) Bearing
AB 201.0 85°20'
BC 240.0 22°40'
CD 194.0 220°0'
DA ? ?
(A) 190.23 m and 242°2'
(B) 87.85 m and 180°28'
(C) 56.71 m and 136°6'
(D) 49.67 m and 44°24'
Solution
Given closed traverse ? SL = SD = 0
Latitude of AB = AB cos q 
= +201cos 85°20'
Departure of AB = AB sin q = +201 sin 85°20'
Line Latitude (m) Departure (m)
+ – + –
AB    16.35 200.33
BC 221.46    92.49
CD 148.6 124.7
Sum 237.81 148.6 292.82 124.7
SL ' = 89.21 SD ' = 168.12
Latitude of DA = –SL' = –89.21 m
Departure of DA = –SD' = –168.12 m
Since both latitude and departure are negative. DA lies in 
SW Quadrant. (IIIQ)
tan q = 
Departure
Latitude
=
168.12
89.21
= 1.88
q = 62°2'
Bearing of DA = S62°2'W = 242°2'
Length of DA = 
Latitude
cos ?
=
° '
89 21
62 2
.
cos( )
=190.23 m.
Hence, the correct answer is option (A).
Part III_Unit 12_Chapter 03.indd   5 5/31/2017   5:01:12 PM