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MEASUREMENT OF
ANGLES AND
DIRECTIONS

MERIDIANS
EXPEDIENT METHODS OF ESTABLISHING
MERIDIANS
UNITS OF ANGULAR MEASUREMENTS
DESIGNATION OF NORTH POINTS
ILLUSTRATIVE PROBLEMS
LESSON
24
MEASUREMENT OF ANGLES AND
DIRECTIONS

OBJECTIVES:
At the end of the lesson the students should be able to:
 Define key terms related to angle and direction measurement
(horizontal angle, vertical angle, bearing, azimuth).
 Practice proper use of compass in measuring angles and
directions.
 Practice proper use of compass in measuring angles and
directions.

MERIDIANS
The direction of a line is usually defined by
the horizontal angle it makes with a fixed
reference line or direction. In surveying, this
is done with reference to meridian which
lies in a vertical plane passing through fixed
point of reference and through the
observer's position.

Four types of
Meridians
True
Magnet
ic
Assume
d
Grid

1. True Meridian
The true meridian is sometimes known as the astronomic or geographic
meridian. It is the generally adapted reference line in surveying practice. This
line passes through the geographic north and south poles of the earth and
the observer's position. Since all true meridians converge at the poles, they
are not parallel to each other.
The direction of true meridian at a survey station Is invariable and any record
of true directions taken remains permanent and unchanged regardless of
time. Lines in most extensive surveys are usually referred to the true
meridian. This meridian Is also used for marking the boundaries of land.

2. Magnetic Meridian.
A magnetic meridian is a fixed line of reference which lies parallel with the
magnetic lines of force of the earth. Its direction is defined by a freely
suspended magnetic needle of a compass held at the observer's position.
Magnetic meridians are not parallel to the true meridians since they converge at
a magnetic pole which Is located some distance away from the true geographic
poles.
Since the location of the magnetic poles changes constantly, the direction of the
magnetic meridian is not fixed. As a line of reference, the magnetic meridian is
employed only on rough surveys where a magnetic compass used in
determining directions.

3. Grid Meridian.
A grid meridian is a fixed line reference parallel to the central meridian of a
system of plane rectangular coordinates. One central meridian, which
coincides with a true meridian, is usually selected and all other meridians are
made parallel to this meridian. In this process, the need to calculate the
convergence of meridians when determining positions of points in the
system is eliminated.
The use of grid meridians is applicable only to plane surveys of limited
extent. In such types of survey, it is assumed that all measurements are all
projected to a horizontal plane and that all meridians are parallel straight
lines.

4. Assumed Meridian.
An assumed meridian is an arbitrarily chosen fixed line of reference which
Is taken for convenience. This meridian is usually the direction from a
survey station to an adjoining station or some well-defined and
permanent point. It Is used only on plane surveys of limited extent since
they are difficult or may be impossible to re-establish if the original
reference points are lost or obliterated.

EXPEDIENT METHODS OF ESTABLISHING MERIDIANS
The following are some expedient methods of determining or establishing meridians.
1. Establishing Magnetic Meridian Py Compass. The magnetic meridian can
be established by setting up the compass over any convenient point and then
sighting a distant object that marks another point on the meridian. For an
accurate determination of the desired meridian, several sights should be
taken during the setup and the compass must be rotated about its vertical
axis and then positioned until the needle reads zero. The mean of the points
thus established is taken as the magnetic meridian. The observations should,
however, be made when the magnetic declination is approximately at its
mean value.

2. Determining True North by Aid of Sun and a Plumb Line.
In a level piece of ground, lean a pole approximately toward the north and rest it
in a crotch made by two sticks (Fig.24-1). Suspend a weight from the end of the
pole so that it nearly touches the ground. About an hour before noon, attach a
string driven directly under the weight and, with a sharpened stick attached to the
other end of the string, describe an arc with a radius equal to the distance from
the peg to the shadow of the tip of the pole. Drive a peg on the arc where the
shadow of the tip of the pole rests. At about an hour after noon, watch the
shadow of the tip as it approaches the eastern side of the arc and drive another
peg where it crosses. By means of a string, find the middle point of the straight
line joining the two pegs. A straight line joining the mid-point and the peg under
the weight will, for all practical purposes, be pointing towards the direction of true
north.

3. Determining True North By the Rising and Setting of the Sun.
From a convenient position or station, observe the rising and setting of the sun on the same day or
at setting on one day and rising the next (Fig. 24-2). Along each direction establish a peg or marker.
Measure the horizontal angle between the two markers then, establish another marker to define
half of the measured angle. The line joining the observation station and the last marker
established should point towards the direction of true north.

4. Determining True North By Polaris.
The big dipper is a useful reference constellation of the northern hemisphere. As a star
group, it is the most familiar and easiest to recognize. It has been so named because of the
distinctive dipper-like pattern formed by seven bright stars (Fig.24-3) . The two stars, Merak
and Dubhe, forming the side of as the dipper which is farthest from the handle are known
the pointer stars. They point towards Polaris which is also Polaris known as the north star,
pole star, or cynosure. lies almost directly above the earth's north pole. When a person
faces Polaris, he is actually facing towards the direction of true north. Polaris is visible the
whole year but only in the northern hemisphere. Aside as a reference for determining
directions, this star can tell a person in the northern hemisphere what latitude he is in. The
observed vertical angle from the horizon to Polaris is approximately the same degree the
Latitude that the observer is from the equator. At equator the vertical angle to Polaris is
zero since the Star is on the horizon. At the north pole, the angle is about 90 degrees since
Polaris is found directly overhead.

5. Determining True South By the Southern Cross.
The southern cross (or crux) is a constellation of the southern hemisphere which serves as a reference
group of stars for determining the location of the earth's south pole. It is composed of four stars formed in
the figure of a cross. An imaginary line joining the two stars forming the longer side of the cross is used to
locate a point directly above the south pole. This reference point is located along the extension of this
imaginary line. Its distance from the lower star (Fig. 24-4) of the cross is about 4.5 times the distance
between the two stars' along the same line.

6. Determining Direction of True North (or South) by a Wrist Watch.
An ordinary wrist watch can be used to determine the approximate direction of true north or south. In the north
temperate zone only the hour hand is pointed toward the sun. A south line can be found midway between the hour
hand and 12 o'clock (Fig. 24-5). The wrist watch may also be used to determine directions in the south temperate
zone. It is done, however, in a different manner. Twelve o'clock is pointed toward the sun, and half-way between 12
o'clock and the hour hand will be the direction towards true north (Fig. 24-6).

UNITS OF ANGULAR MEASUREMENT
1. The Degree.
The sexagesimal system is used in which the circumference of a
circle is divided into 360 parts or degrees. The angle of one
degree is defined as the angle which requires 1/360 of the
rotation needed to obtain one complete revolution. The basic
unit is the degree, which is further subdivided into 60 minutes,
and the minute is sub-divided into 60 seconds. The ' , ' and " are
used to denote degrees, minutes, and seconds, respectively.
Thus an angle 26 degrees, 32 minutes, and 15 seconds may be
written as 26' 32'15". If decimal parts of degrees is desired the
above value may be written as 26.5375 degrees. This system is
used extensively in surveying practice.

2. The Grad.
The grad is the unit of measure in the centesimal system. In this
system the circumference of a circle is divided into 400 parts called
grads. The grad is subdivided into 100 centesimal minutes and a
centesimal minute is further subdivided into 100 centesimal
seconds.
The symbols g, c and cc are used to denote grads, centesimal
minutes, and centesimal seconds, respectively. It will be noted
that 200 grads is equal to 180 degrees.
An angle may be expressed as 235. 26180 where the first pair of
digits to the right of the decimal point represents centigrads and
the last pair of digits farther to the right of the decimal point
represents the decimilligrads. The preceding value may also be
written as 235^g 26^c 18^cc

3. The Mil.
The circumference is divided into 6400 parts
called mils, or 1600 mils is equal to 90
degrees. The mil will subtend very nearly
one linear unit in a distance of 1000 such
units. It is commonly used in military
operations as in fire direction of artillery
units.

4. The Radian.
The radian is another measure of angles used frequently for a
host of calculations. One radian is defined as the angle
subtended at the center of a circle by an arc length exactly
equal to the radius of the circle. One radian equals 180/TT or
approximately 57.2958 degrees and, one degree equals TT /180
for approximately 0.0174533 radians. The radian is sometimes
referred to as the natural unit of angle because there is no
arbitrary number in its definition. It is used in computations
such as determining the length of circular arcs and where high
speed electronic digital computers are used.

DESIGNATION OF NORTH POINTS
There is always a starting or reference point to define the directions. Map
users are primarily concerned with north point for the determination of
directions and the following are the commonly used reference points.
1. True North - is the north point of the true meridian. In maps and
sketches, it is portrayed in the direction of the actual location of the
earth's north geographic pole and is always shown along a vertical
line. It is symbolized by a star, an asterisk or the letters TN. (fig24-9a)

2. Magnetic North - a north point that is established by means of a magnetized
compass needle when there are no local attractions affecting it. At any point on
the earth's surface its direction is indicated by the direction of the magnetic lines
of force passing through the point at a Magnetic north may be located either east
particular time. or west of true north. The point is usually symbolized by a half
arrowhead or the letters MN (Fig. 24-9b).
3. Grid North - a north point which is established by central lines on a map which
are parallel to a selected meridian. It may coincide with lines directed toward
true north. Grid north may be symbolized by a full arrowhead or the letters GN or
Y (Fig. 24-9c).
4. Assumed North - is used to portray the location of any arbitrarily chosen north
point. It may be symbolized by a small blackened circle or the letters

DIRECTION OF LINES
INTERIOR ANGLES
DEFLECTION ANGLES
ANGLES TO THE RIGHT
BEARINGS
FORWARD AND BACK BEARINGS
AZIMUTHS
FORWARD AND BACK AZIMUTHS
LESSON
25
MEASUREMENT OF ANGLES AND DIRECTIONS

The direction of a line is defined as the horizontal angle the line makes with an
established line of reference. There are various kinds of angles which can be used to
describe the direction of lines. In surveying practice, directions may be defined by
means of: interior angles, deflection angles, angles to the right, bearings, and
azimuths.
Angles are measured or laid off directly in the field by using devices such as a
compass, transit, theodolite, sextant, or by plane table and alidade. The steel tape
may also be used to lay off or measure angles. These angular quantities are said to be
observed when obtained directly in the field with a measuring instrument and
calculated when obtained indirectly by computations. Angles are computed by means
of their relationship to known quantities in a triangle or other geometric figures.
DIRECTION OF LINES

INTERIOR ANGLES The angles between adjacent lines in a closed polygon are called
interior angles. In Figure 25-1, the interior angles are Фа Фb , Фс г Фd, and Фe . These
angles may be measured clockwise or counterclockwise. When the value of an interior
angle is greater than 180 degrees it is referred to as a re-entrant angle. One such
example is the interior angle at station E or Фe. It should be remembered that for any
closed polygon the sum of the interior angles is equal to (n-2)180 degrees, where n is
the number of sides. For the polygon shown in Figure 25-1, the sum of the interior
angles is (5-2)180 degrees or 540 degrees.
Exterior angles are located outside a closed polygon and are referred to as
explements of interior angles. An explement is the difference between 360 degrees
and any one angle. These angles are often measured in surveying work and used as a
check, since the sum of the interior and exterior angles at any station or point must
equal to 360 degrees. In Figure 25-2.

DEFLECTION OF LINES
The angle between a line and the prolongation of the preceding line is called a deflection angle. It may be turned to
the right (clockwise) or to the left (counterclockwise) and it is always necessary to append the letters R or L to the
numerical value to define the direction in which the angle has been turned. Right deflections are considered to have
signs opposite to left deflections. Usually, a positive sign is used to define a deflection angle to the right and a negative
sign for deflection angles to the left.
In Figure 25-3, the deflection angles at stations B, C, and D are W. (R), w.(L), and w.(R), respectively. These angles may
have values between 0 and 180 degrees, but often they are not used for angles greater than 90 degrees. In any closed
polygon the algebraic sum of the deflection angles should always equal to 360 degrees.

ANGLES TO THE RIGHT
Angles to the right are measured clockwise from the preceding line to the succeeding line.
In figure 25-4, the angles to the right at stations B, C, and D. These angles are also referred
to as azimuths from back line.

BEARINGS
The direction of a line may be described by giving its bearing. The bearing of a
line is the acute horizontal angle between the reference meridian and the line. A
quadrantal system (Fig. 25-5) is used to specify bearings such that a line may fall
under one of the following quadrants: NE, SE, NW, and SW. Each quadrant is
numbered from 0 to 90 degrees from either the north or south end of the
meridian to the east or west end of the reference angles parallel (or the E-W
Line). The fact that bearing angles never exceed 90 degrees is an advantage
when extracting values of their trigonometric functions for use in computations.

BEARINGS
Either the letters N or S precedes the bearing angle and the letters E or W follows the
indicated value of the angle. It is never done the other way around. Therefore, to locate
a line it is always necessary to know the directional quadrant in which it lies as well as
the angle it makes with the reference meridian. The line could lie in any of the four
quadrants if only the bearing angle of the line is known.

BEARINGS
Bearings may also be designated in a different manner when the direction of a line lies in
the same direction as the reference meridian or reference parallel. If the line lies parallel
to the meridian and south, it is written as due south; if perpendicular to the meridian and
east, it is written as due east. In figure 25-6, the bearing of lines originating from point p
are given.

FORWARD AND BACK
BEARINGS
Using the quadrantal system, any line on the surface of the earth may be
defined by two directions which differ from each other by exactly 180
degrees. The direction will depend on which end the line is observed. When
the bearing of a line is observed in the direction in which the survey
progresses, it is referred to as a forward bearing, if this bearing of the same
line is observed in an opposite direction it is called the back bearing.

FORWARD AND BACK BEARINGS
In Figure 25-7, assume a compass is set up successively at stations A, B, C, D, and E, and bearings read on lines AB,
BA, BC, CB, CD, DC, DE, and ED. The observed bearings of lines AB, BC, CD, and DE are called forward bearings; those
of BA, CB, DC, and ED are back bearings. From the illustrated directions given in Figure 25-7, it can be readily seen
that back bearings can be obtained from the forward bearings by simply changing the letter N to S and also changing
E to W, and vice versa.

AZIMUTHS
Another common method used in designating the direction of a line is by the use of
azimuths. The azimuth of a line is its direction as given by the angle between the meridian
and the line measured in a clockwise direction from either the north or south branch of the
meridian. Azimuths are usually preferred over bearings by most surveyors because they are
more convenient to work with such as in computing traverse data by electronic digital
computers.
The azimuth of a line may range from 0 to 360 degrees and letters are not required to identify
quadrants. For any particular survey the direction of zero azimuth is either always north or
always south. Some surveyors reckon azimuths from the south and some from the north
branch of whatever meridian is selected as a reference. Usually a particular agency or
organization will consistently use one or the other. Since both the north and south branches
of the meridian are used, it is important to always specify and record which branch is used
whenever azimuths are recorded.

AZIMUTH
In practice, azimuths are generally
measured from the north branch of the
reference meridian for ordinary plane
surveys. For large scale geodetic
surveys and in astronomical
observations azimuths are measured
from the south branch of the meridian.
Figure 25-8 shows different lines whose
azimuths are measured from the north
branch of the reference meridian.
Azimuths measured from the south
branch of the meridian are shown in
Figure 25-9.

Any line established on the earth's surface has two azimuths - a forward azimuth and a back
azimuth. Depending on which end of the line is considered, these directions differ by 180
degrees from each other since the back azimuth is the exact reverse of the forward
azimuth. To determine the back azimuth when the forward azimuth is known, the following
rules are used:
RULE 1: If the forward azimuth of the line is greater than 180 deg. subtract
180 deg. to obtain the back azimuth.
RULE 2: When the forward azimuth of the line is less than 180 deg. add 180
deg. to determine the back azimuth.

Shown in Figure 25-10 are four
successive lines whose azimuths have
been observed. Tabulated immediately
below the figure are the observed
forward and back azimuths (reckoned
from south) of lines AB, BC, CD, and DE.
The tabulation also shows the
calculated forward and back azimuths
of each line as reckoned from the north
branch of the reference meridian. By
applying Rules 1 and 2, the student
should be able to determine how the
tabulated back azimuths have been
determined.

ACTIVITY
The class will be divided into 3 groups. Each group will
Draw a triangle on paper, measure the interior angles
using a protractor, check that the total equals 180°,
then identify the directions of each side using bearings
(e.g., N30°E), and label the directions accordingly while
ensuring your work is neat and organized.

Quiz
Get ¼ sheet of yellow pad paper
Read carefully and answer the following questions.

1. The direction of a line is usually defined by the horizontal angle it makes with a fixed reference
line or direction.
a) MERIDIAN
b) Azimuth
c) Angle
d) Degree
2. It is defined as the horizontal angle the line makes with an established line of reference.
a) Azimuth
b) Degree
c) Direction of lines
d) Meridian
3. The angles between adjacent lines in a closed polygon.
a) Azimuth
b) Degree
c) Interior angles
d) Exterior angles

4. located outside a closed polygon and are referred to as explements of interior angles
a) Azimuth
b) Degree
c) Interior angles
d) Exterior angles
a. The sexagesimal system is used in which the circumference of a circle is divided into 360 parts
a) Azimuth
b) Degree
d) Meridian
6. These are generally measured from the north branch of the reference meridian for ordinary
plane surveys.
a) Meridian
b) Degree
d) Azimuth

7-10. What are the 4 types of meridian

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