Angles
1
Expert vs. Non-expert
“The non-expert fails most of the time because his success
depends upon meeting conditions which coincide with a fixed,
and usually limited range of mentally-cataloged techniques,
whereas the expert, because of his fundamental understanding
of what he is trying to achieve, thinks more in terms of how and
why, than of what; and thus is able to devise specific techniques
in response to the demands of specific conditions. Through
understanding, he achieves a kind of infinite flexibility.”
Brian Clarke
2
Introduction
• Angle: The rate two lines or planes diverge from a common
point.
• Parts of an angle
– Backsight
– Vertex
– Foresight
3
Angle Units
• Several different units can be used to measure angles.
• This class uses two.
– Decimal Degrees (DD)
– Degrees Minutes Seconds (DMS)
4
Angle Units-DD
• DD expresses any part of an angle less than a whole degree as
a decimal.
– 108.24o
• Electronic instruments such as total stations and GPS can
output angles in DD.
• Angles in DD is the system of choice today because it is the
easiest form to use with calculators and computer software.
5
Angle Units-DMS
• DMS is the angle measuring method used on most mechanical
instruments.
– 108o 23’ 40”
• In the DMS system there are 60 minutes in each degree and 60
seconds in each minute.
• Because both systems are still used, it is useful to know how to
convert from one to the other.
6
DMS to DD
• Many calculators have a DMS to DD and DD to DMS conversion
key.
• It will save a lot of time and reduce mistakes if you learn how to
do these conversions on a calculator.
• If you cannot do it on a calculator, then you must learn how to
do it manually.
To convert from a DMS angle to a DD angle the minutes and seconds
must be converted to a fraction. The fractions are reduced to decimal
equivalents and then the parts are added.
7
DMS to DD-cont.
Example: Convert 120o 34’ 45” to DD
34
45
+
60
3, 600
 120  0.566... + 0.0125
120o 34' 45" = 120 +
o
= 120.5791... or 120.58

8
DD to DMS
The manual method from DD to DMS follows the same math
principles.
The decimal part of the angle must be converted to minutes and
seconds.
Example: convert the
angle 45.349o to DMS
45.349 = 45 o
0.349 x 60 = 20.94 = 20'
0.94 x 60 = 56.4"
Answer 45 o 20' 56"

9
Adding & Subtracting Angles in DMS
Occasionally when using mechanical instruments it is necessary to
add and subtract angles using DMS.
The addition and subtraction principles are the same, except units
of 60 are carried or subtracted instead of units of 10.
10
Adding Angles in DMS
Example: Add the angles 20o 45’ 27” and 30o 24’ 35”
20o 45Õ 27"
+ 30o 24Õ35"
50o 69Õ 62"
In is not proper to leave an angle
measurement with more that 60 minutes or
seconds.
The answer must be reduced.
First step is to reduce the seconds,
50 o 69' 62"
= 50 o 70' 02"
The next step is to reduce the minutes,
50 o 70' 02"
= 51o 10' 02"

if needed.
if needed.
The answer is :
51o 10' 02"


11
Subtracting Angles In DMS
Subtraction follows the same principles.
Example: Subtract 40o 18’ 50” from 120o 15’ 45”
120 o 15' 45"
-
40 o 18' 50"
• 45 - 50 and 15 - 18 would result in
negative numbers.
• 120o 15’ 45” must be converted to:
119 o 74' 105"
119 o 74' 105"
-

40 o 18'
79
o
56'
The answer is:
50"
55"


79 o
56'
55"
12
Types of Angles
• Surveying uses at least six (6) types of angles.
• It is important to record the type of angle that was
measured
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Angles-Six Types
1. Interior
2. Exterior
3. Angle to right
4. Angle to Left
5. Right Deflection
6. Left Deflection
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Azimuth--Bearing
•
•
Direction of movement is usually recorded as an azimuth or a bearing.
Azimuth:
– An angle measured from north.
– North can be magnetic north or geographical north.
– Important to record which is used.
– Azimuth angles are either right turned (+) or left turned (-).
•
Bearing
– Bearings are relative angles.
– They are used to express the direction of travel from one point to the next.
– Bearings are based on the quadrant N, S, E, W.
– Bearing must always start and end with a compass direction.
15
Azimuths
• Note: a right azimuth is (+)
plus and a left azimuth is () minus.
• Left is not a negative
angle, the (-) is used to
denote direction.
16
Bearings
• Bearings are based on quadrants.
• Bearing angles are always measured from a N-S Line.
• The bearing always starts with north (N) or south (S) and ends
with east (E) or west (W).
• Example:
17
Azimuths to Bearings
•
•
•
Occasionally it is necessary to convert
from azimuths to bearings.
Conversions are based on + or - azimuth
and the number of degrees in a circle,
360.
Examples:
– Determine the bearing for an azimuth
of 53.0o.
– Add NESW and determine angle.
18
Azimuth to Bearing #2
• Determine the bearing for an azimuth of 135.9o.
– Draw azimuth.
– Draw NESW and determine bearing.
19
Azimuth to Bearing #3
• Determine the bearing for an azimuth of
+275.6o.
20
Bearing to Azimuth
• Because bearings are relative to
the direction of travel, there are
two possible azimuths for each
bearing, (+) & (-).
21
Bearing to Azimuth-example
• Example: Determine the + azimuth for a bearing of S32.9E.
– Step one: draw the NESW and the bearing angle.
– Step Two: determine the right turned azimuth for the bearing.
– What would be the - azimuth for the bearing?
22
Measuring Angles
• There are two methods for measuring existing or laying out new
angles.
– Indirect
– Direct
• Indirect methods measure and lay out angles by utilizing
equipment that can not measure angles directly.
• Direct measurement and lay out of angles is accomplished by
instruments with angle scales.
23
Indirect Angle Measurement
• The common processes for indirect measurement of angles are:
–
–
–
–
Chord
Simplified Chord
3-4-5
Tape sine
24
Chord Method
• A chord is a line segment joining two
parts on a circle.
• The chord indirect method uses this
principle to lay out a 90o angle.
– Step one: establish a base line and the
vertex of the angle such that A:C = B:C.
– Step two: Select a length longer than A:C
and scribe two arcs using stations A and
B as the centers.
Step three: Establish a station where
the two arcs cross, D.
Step four: A line drawn from C
through D will establish a 90o angle
with the base line at C.
25
Chord Method--cont.
• Advantages
– Doesn’t use surveying
instrument.
– Can be accomplished with a
string or stick.
– Doesn’t require math
• Disadvantages
– Limited to 90o.
– Can not be used to measure
existing angle.
– Must be able to extend the
base line past the vertex.
– Can be difficult to establish
the arcs on some surfaces.
26
Simplified Chord Method
This method utilizes the principles of the chord method and a piece of string.
– Step one: Obtain a
piece of string 10 to
12, or more, feet
long.
– Step two: Tie a knot
in each end and
either a knot or loop
at the mid point of the
end knots.
– Step three: Establish – Step four: Attach or
the same type of base
hold one end knot on
line by folding the
station A and the other
string into fourths
end knot on station B.
(1/4’s). Use this length – Step Five: Move the
to lay out the distances
center knot away from
A:C and B:C.
the base line in the
desired direction of the
90o line.
– Step Six: Move the
center knot until there is
equal tension on each
side.
– Step Seven: Mark the
location of the center
knot. A line drawn from
the vertex through this
point will establish a 90o
line with the base line.
27
3-4-5 Method
• The 3-4-5 method is based on the principles of triangles.
• Any triangle with a ration of the lengths of the sides equal to 3, 4
& 5 will have one right (90o) angle.
28
3-4-5 Method--cont.
• This method can be used with a single measuring tape
and three people.
• It can also be completed with two measuring tapes
and one person.
29
3-4-5 Method--One Tape
• Step one: Establish the base line and
the vertex of the angle. Use a
reference of A for the vertex.
• Step two: Mark station B. Use either 3
or 4 or a multiple of either one.
• Step three: Because steel tapes will be
damaged if they are bent sharp, form a
2 to 5 foot loop in the tape at B and
extend the tape toward the estimated
position of C.
• With a two (2) foot loop, the 6 & 8 foot
marks should be held a station B.
• Distance B:C is a multiple of 5. In this
case 10 should be used.
30
3-4-5 Method--One Tape
• Step four: Make another loop using foot
mark 18 and 20 at the estimated
position of C and extend chain to A.
• Hold the 28 and zero foot marks together
at station A.
• Stretch both legs until equal tension and
mark station C.
• Stations A, B & C form a right angle and
station A.
31
3-4-5 Two Tapes
• Step one: Establish the base line and the vertex of the angle.
Use a reference of A for the vertex.
• Step two: Measure and mark either 3 or 4, or a multiple, starting
at the vertex and mark this station as B.
• Step three: Attach one tape to station
A and one tape to station B.
– If the vertex is at A, stretch the tape
attached to A out to 8 feet and the tape
attached to B to 10 feet.
– Hold the 8 and 10 foot marks together
and place a pin at that point and
reference it as station C.
– A line extended from station A through
station C will form a 90o angle with the
base line.
32
Advantages & Disadvantages
of 3-4-5
• Advantages
– Base line does not need to
extend past the vertex.
– More accurate.
• Disadvantages
– The single tape method
requires three people.
– Can only be used to lay out
90o angles.
33
Tape-sine
• As the name implies, the tape sine method requires the use of
both a tape measure and the sine trigometric function.
• Advantages
– Can measure and layout
angles.
– Can measure and layout
any angle between 0o and
90o.
• Disadvantages
– Requires the use of a
calculator or trig tables.
– Mathematical procedure.
34
Measuring An Existing Angle With Tape-sine
Example: Determine the angle of the side walks in the illustration.
• Step one: establish two
lines of equal length along
each path of the angle,
starting at the vertex and
label them as A, B & C.
– The calculations are
easier if this number is
10 or 100 feet, but any
length will work.
35
Measuring An Existing Angle With Tape-sine--cont.
• Step two:
– Measure and record the
distance from A to C.
• Step three:
– Divide distance A:C in
half and use this point
with B to form two right
triangles.
36
Measuring An Existing Angle With Tape-sine--cont.
• Step four:
– Use 1/2 of the A:C
distance and the 100 foot
side to calculate the ABC’
angle for one of the
triangles.
• The angle of ABC is the
calculated angle times 2.
55 ft
Opposite
2 = 0.275
Sin  =
=
Hypotenouse
100 ft
 = 15.96
15.96 x 2

= 31.9 o
37
Establishing An Angle Using Tape-Sine
Example: You want to layout retaining wall that that has an eighty
(80) degree angle with an existing retaining wall.
Lay out the angle for the retaining wall using the tape-sine method.
38
Establishing An Angle Using Tape-Sine--cont.
Step one: Lay out a distance along the existing wall.
Whole numbers are best.
39
Establishing An Angle Using Tape-Sine--cont.
Step two: Calculate the distance from station B to C’ using half of
the desired angle.
Sin  =
Opposite
Hy potonouse
Opposite = Sin  x Hy potonouse
= Sin 40 x 50 f t
= 32.14 f t
The distance B:Cis twice this distance.
32.14 x 2 = 64.28 ft
40
Establishing An Angle Using Tape-Sine--cont.
Station C is located by stretching one tape 50 feet from station A
and another tape 64.3 feet from station B.
The point where these two distances cross is station C.
41
Direct Methods of Measuring Angles
42
Introduction
• Direct methods of measuring angles involves the use of
surveying equipment with angle scales.
• The operator must understand how to use each type of
instrument.
• Mechanical instrument use Vernier scales---must know how to
read Vernier scales accurately to obtain useable data.
43
Reading Single Vernier Scale
• Determine whole units
• Determine the first
graduations.
• Determine the next
graduations--if any.
• Read the first two values at
the zero point of the Vernier
scale.
• Find line of coincidence on Vernier scale.
• Added Vernier reading to the smallest reading on the main scale.
Reading: 32o 14’
44
Reading Double Vernier Scale
• Some mechanical surveying instruments use a double scale
so angles can be directly read when turn to the left or right.
• Recommended to determine the least count of the instrument
before attempting to read the Vernier.
–
–
–
–
–
–
In this example, the numbers on the main scale are degrees.
Each degree is subdivided into 1/3’s
60’/3 = 20’
The Vernier scale range is from 0 to 20 = minutes
There are 19 lines between 0 and 20 on the Vernier scale
The least count is one minute.
45
Reading Double Vernier Scales
• The right turned angle is: 335
o
+ 17Õ = 335o 17Õ

•
Remember the Vernier scale moves and to read the half of the Vernier scale
that is leading in the direction of rotation.
• The Left turned angle is:
24o + 40' +7' = 24o 47'

46
Vernier Movement
• Left and right
turned refers to the
rotation of the
telescope.
• Right turned is
clockwise (CW).
– Vernier appears
to turn left, but it
is clockwise also
• Left turned is counter clockwise (CCW).
–Vernier appears to turn right.
47
Reading Double Vernier Scales--cont.
• The left turned angle is: 44o + 40' +2' 30" = 44o 42' 30"

• The right turned angle is: 310 o + 50 + 20' +13' = 315 o 33'

48
Direct Reading Instruments
49
Automatic Level
• Only measures horizontal
angles.
• The precision is 1.0 degree.
• Has only a single angle scale.
• The automatic level uses a
friction movement. There are
no locks, but there is a tangent
screw.
• Alignment is by two pins on top
of instrument.
• Vertical line of backsight and foresight must be established with
a range pole or rod.
• Rod must be held plumb to obtain accurate angle reading.
50
Turning Angle With
Automatic Level
1. Use plumb bob to center instrument
over vertex.
2. Level instrument.
3. Adjust eye piece
4. Focus telescope
5. Align telescope on backsight station.
6. Rotate angle scale to zero.
7. Turn angle (rotate instrument to
foresight station).
8. Read and record.
51
Dumpy Level
•
•
•
Only measures horizontal angles.
The precision is 5 minutes.
Uses a quadrant horizontal angle
scale.
• Has locks on the movement and a
tangent screw.
• Alignment is sighting top of
instrument.
• Vertical line of backsight
and foresight must be
established with a range
pole or rod.
52
Transit
• Transits have more capabilities and higher
precision than most mechanical instruments.
–
–
–
–
Horizontal angles
Vertical angles
Azimuth angles
Bearings
1927 Transit
• Because of the greater capabilities, they are
more mechanical complicated.
– Double angle scales and double Verniers.
– Upper and lower plates for turning horizontal
angles.
• Precision is only exceeded by Theodolite
– K&E transits are 20 seconds.
53
Transit--Horizontal Angles
(Setting up and zero setting)
• Drop the plumb bob through the head of the tripod and align
tripod over the vertex.
• Align the transit over the vertex using the plumb bob.
• Level the instrument.
•Loosen the upper plate and rotate the “A” angle scale to zero.
–Use the tangent screw and eye loop to be precise.
•Tighten the upper plate.
54
Transit--Horizontal Angles
(Aligning on backsight)
• Loosen the lower plate and rotate the telescope until it is aligned
on the backsight.
• Adjust the eye
piece.
• Focus the
telescope on the
backsight.
– Depress the vertical movement until the crosshairs can be set on the
backsight.
– Lock the lower plate.
– Release the upper plate and turn the angle.
– Lock the upper plate and read the angle.
• Remember to record + or -.
55
Transit--Horizontal Angle
(Azimuth Angle)
•
•
•
•
•
Set up instrument
Level instrument
Zero set “A” scale.
Release the lower movement.
Release the compass needle and
rotate the telescope horizontally until
the compass reads zero degrees.
–
•
•
•
Lock lower movement.
Loosen upper movement.
Turn the desired angle.
–
•
•
Transit compasses are very sensitive
to magnetic fields. Heavy metal watch
bands or jewelry can cause the needle
to deflect.
Adjust eye piece and focus as
necessary.
Lock upper movement.
Read the angle.
–
Remember to record + or -.
56
Transit--Turning Vertical Angles
•
Set up transit on tripod insuring plumb
over vertex and level instrument.
• Note: the K&E transit uses horizontal
zero.
• Rotate the telescope until vertical
crosshair is aligned with angle station.
• Depress the telescope until the horizontal
crosshair is on the backsight of the
angle.
• Read the vertical angle scale.
• Elevate the telescope until the horizontal
crosshair is on the foresight of the angle.
• Read the vertical angle scale.
• Add the two readings to obtain the
angle.
Note: the vertical scale has a precision of 15 minutes.
57
Questions
58