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.

– 108.24

o • 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.

– 108 o 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 120 o 34’ 45” to DD 120 o 34' 45" = 120 +  120 34 + 45 60 3, 600  0.566... + 0.0125

= 120.5791... or 120.58

o 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 45.349

o to DMS

45.349 = 45

o

0.349 x 60 0.94 x 60 = 20.94 = 20' = 56.4"

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 20 o 45’ 27” and 30 o 24’ 35”

20

o

45’ 27" + 30

o

24’ 35" 50

o

69’ 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, o  o 10' 02" if needed.

The answ er is :

if needed. o

10' 02"

 11



Subtracting Angles In DMS

 Subtraction follows the same principles.

Example: Subtract 40 o 18’ 50” from 120 o 15’ 45” 120 o 15' 45" - 40 o 18' 50" 119 o 74' 105" - 40 o 18' 50" 79 o 56' 55" • • 45 - 50 and 15 - 18 would result in negative numbers.

120 o 15’ 45” must be converted to: o 74' 105" 

The answer is:

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 13

Angles-Six Types

1. Interior 2. Exterior 3. Angle to right 4. Angle to Left 5. Right Deflection 6. Left Deflection 14

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 53.0

o .

– Add NESW and determine angle.

18

Azimuth to Bearing #2

• Determine the bearing for an azimuth of 135.9

o .

– Draw azimuth.

– Draw NESW and determine bearing.

19

Azimuth to Bearing #3

• Determine the bearing for an azimuth of +275.6

o .

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 90 B as the centers.

o 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 Step three: Establish a station where the two arcs cross, D.

Step four: A line drawn from C through D will establish a 90 o 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 90 o .

– 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 the same type of base line by folding the string into fourths (1/4’s). Use this length to lay out the distances A:C and B:C.

– Step four: Attach or hold one end knot on station A and the other end knot on station B.

– Step Five: Move the center knot away from the base line in the desired direction of the 90 o 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 90 o 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 (90 o ) 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 90 o 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 90 o 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 0 o and 90 o .

• 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.

Sin  = Opposite Hypotenouse = 55 ft 2 100 ft = 0.275

 = 15.96 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 = Hypotonouse Opposite = Sin  x Hypotonouse = Sin 40 x 50 ft = 32.14 ft 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: 32 o 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: o

+ 17’= 335

o

17’

•  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: o + 40'+7'= 24 o 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: o + 40'+2' 30" = 44 o 42' 30"  • The right turned angle is: o + 5 0 + 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 • 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. 1927 Transit 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.

– Transit compasses are very sensitive to magnetic fields. Heavy metal watch bands or jewelry can cause the needle to deflect.

Lock lower movement.

Loosen upper movement.

Turn the desired angle.

– 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