ANGLES AND DIRECTIONS

RELATED DEFINITIONS:

1. BEARING – the bearing of a line is the acute horizontal angle between a

reference meridian and a line. The angle is measured from either the north or
south toward the east or west, true bearings are measured from the true
meridian, magnetic bearings from the local magnetic meridian, assumed
bearings from any adopted meridian, and grid bearings from the appropriate
grid meridian.
2. AZIMUTH – horizontal angles which are measured clockwise from the north
or south end of the meridian. Azimuths may be true, magnetic, grid, or
assumed, depending on the meridian used.
3. TRAVERSE – a series of distances and angles, or distances and bearings, or
distances and azimuths, connecting successive points.
4. OPEN TRAVERSE – a series of lines of known lengths and direction which
are continuous but do not return to the starting point.
5. CLOSED TRAVERSE – a series of lines known lengths and directions which
return or close back to the point of beginning or at points whose positions
have been fixed by other surveys of higher precision.
6. LOCAL ATTRACTION – the amount of deviation or deflection from the normal
pointing of the magnetic needle towards magnetic north due to local
influences such as the proximity of steel, iron, magnetic ores. And direct
electric current.
7. BEST LINE – a line in a traverse which appears to be free from local
attraction. It is also on this line where the adjustment of the remaining
traverse lines affected by local attraction is started.
8. ANGULAR ERROR OF CLOSURE – the angular error of closure for a closed
traverse is the difference between the sum of the computed interior angles
and the quantity (n-2)x180⁰, where “n” is the number of sides in the traverse.

EXAMPLES

Example 1
For the traverse shown in Figure 1, the bearing of side AB is given as well as the interior
angles at B and C. Compute the north azimuths and the bearings of sides BC and CD.

FIGURE 1
Computing azimuth and
bearing for side BC

Deflection A @ C = 180°-76°44’ = 103°16’R

Azimuth of CD = 114°52’+103°16’

= 218°08’

Bearing of CD = 538°08’W

Example 2
The magnetic bearing of line AB was recorded as S43°30'E in 1988. If the magnetic
declination was 2°00’E, what is the astronomic bearing of the line? If the declination is
now 3°00'W, what is the magnetic bearing of the line today?
Solution

FIGURE 2

Astronomic bearing

of side AB.
Magnetic bearing

of side AB

PROBLEMS

a) Convert 51.3165g to sexagesimal units. (Ans.: 46° 11 '05")

b) Convert 67°16'45" to centesimal units (Ans.: 74.7546g)

A circular arc has a radius of 620.00 ft and a central angle of 41°15'30".

Determine the central angle in radians and the arc length.

Three lines have the following north azimuths.' 141°16', 217°23' and 34 °37'.

What are their bearings? (Ans.: S38°44'E, S37°23' W, N48°23'W)

Determine the north azimuths for sides AB, BC, and CD in the accompanying sketch for

which the bearings are given.

Calculate the north azimuth for sides OA, OB, OC, and OD in the accompanying figure.

(Ans.: 17°22'16"; OB = 109°41'14")

Find the bearings of sides BC and CD in the accompanying figure.

Compute the bearings of sides BC and CD in the accompanying figure.

(Ans.: S17°32'E; CD = N85°10'W)

What are the bearings of sides CD, DE, EA,

and AB in the accompanying figure?

Determine the angles AOB, BOC, and DOA for the figure of Problem 9-5

(Ans.: AOB= 92°18'58"; BOC = 81°27’38”)

Compute the value of the interior angles at B and C for the figures shown.
In Problems with figure shown below, compute all of the interior angles for each of the

figure shown.

Compute the deflection angles for the traverse of Problem 9-12.

 (Ans.: B = 146°06'33"R; C = 58°25’02” R)

Compute the deflection angles for the traverse of Problem 9-14.

From the data given, compute the missing bearings.

1-2 =_________

2-3 =_________

3-4 = N8°10'00"W

4-1 = _________

Interior    at 1 = 51°16'00"

Interior    at 2 = 36°22'00"

Interior    at 4 = 221°37’36"

(Ans.: 1—2 = N78°56'04"; 2-3 = S42°34'04” W)

From the data given. Compute the missing bearings

1-2 = N25°49’E

2-3 = _______

3-4 = S35°18’W

4-5 = _______

5-6 = N81°11’W

6-1 = ________
Interior at 1 = 114°06’

Interior at 2 = 79°54’

Interior at 4 = 242°38’

For the accompanying figure, compute the following:

a. Deflection angle at B. (ans.: 72°28’R)

b. Interior angle at B. (ans.: 107°32’)

c. Bearing of the line CD. (ans.: S26°40’E)

d. North azimuth of DA. (ans.:289°18’)

For the figure shown, compute the following:

a. Bearing of line AB

b. Interior angle at C

c. North azimuth of line DE

d. Deflection angle at B

For the figure shown, compute the following:

a Deflection angle at B. (Ans.: 93°51 'R)

b Bearing of CD. (Ans.: S61°27'W)

c North azimuth of DE. (Ans.: 159° 54')

d Interior angle at E. (Ans.: 57°12')

e Exterior angle at F (Ans.: 244° 52')

The following values are deflection angles for a closed traverse: A = 109°18'R, 84°27'R,

113°43'R, D = 80°22'L, and E = 132°54'R. If the bearing of side CD is S24°16'W,

compute the bearings of the other sides.

The magnetic north azimuth of a line is 125°43' while the magnetic declination is 8°30'E.

What is the astronomic azimuth of the line? (Ans.: 134°13')

At a given place the magnetic bearings of two lines are N42°41'E and S58°35'E. If the

magnetic declination is 4°50'W, what are the astronomic bearings of the lines?

The magnetic bearings of two lines are N 14030'E and S85°30'E. If the magnetic

declination is 5°40'E, what are the astronomic bearings of the lines?

(Ans.: N20°10'E; S79°50'E)

The astronomic bearings of two lines are N85°44'E AND S43°38'W. Compute their

magnetic bearings if the magnetic declination is 4°40'E.

Change the following astronomic bearings to magnetic bearings for a 3°45'W magnetic

declination: N5°32'W, N16°32'E and S88°22'E.

(Ans.: N1°47' W, N20°17'E, S84°37'E)

The magnetic north azimuth of a line was 134°30' in 1890 when the magnetic

declination was 6°30'E. If the magnetic declination is now 3°20'W determine the

astronomic azimuth of the line and its magnetic azimuth today.

In 1860 the magnetic bearing of a line was S81°30'E and the magnetic declinations

3°15'W. Compute the magnetic bearing of this line today if the magnetic declination is

now 4°30'E„ What is the astronomic bearing of this line?

(Ans.: Astronomic = S84°45'E, magnetic = S89°15'E)

From the information given in Problems 9-30 to 9-33, determine the astronomic bearing

of each line and its magnetic bearing today.

Magnetic bearing Magnetic Magnetic

in 1905 declination in declination today

1905
9-30 N41°30’W 4°00’W 3°30’E
9-31 S85°45’ 5°30’E 4°15’W
9-32 N5°15’E 4°15’W 9°15’E
9-33 N6°30’E 7°30’W 4°45’W
(ans.: 9-31: astronomic = N88°45’W; magnetic = N84°30’W)

(ans.: 9-33: astronomic = N1°00’W; magnetic = N3°45’E)

1. Explain the difference between turning points and bench marks

2. From the given data, complete and check the level notes shown.

Station BS HI FS Elevation

BM1 3.10 100.00

 TP1 6.44 9.82

TP2 8.76 6.96

TP3 7.12 3.27

BM2 10.44

Station BS HI FS Elevation

BM1 7.432 864.384

TP1 7.121 1.862

TP2 6.946 6.733

TP3 5.397 6.491

TP4 4.312 4.682

BM2 5.111 (ans.: BM2 =

870.713)

Station BS HI FS Elevation

BM1 5.462 642.230

TP1 6.317 4.792

TP2 3.339 3.102

BM2 6.217 1.964

TP3 7.593 3.326

BM3 5.288
Station BS HI FS Elevation

BM11 8.66 1584.61

TP1 4.41 1.96

4.24

3.18

TP2 5.36 6.96

5.41

5.11

TP3 7.42 10.02

BM12 8.64 (ans.: BM12 =

1582.88)