3.0 Introduction 

What is a horizontal angle? 

1. In topography, the angle made by two ground lines is measured horizontally, and is called a horizontal angle. You may replace these ground lines by two lines of sight AB and AC. These lines of sight are directed from your eyes, which form the summit A of the angle BAC, towards permanent landmarks such as a rock, a tree, a termite mound, a telephone pole or the corner of a building. 
The lines of sight in angle BAC 

Expressing horizontal angles 

2. Horizontal angles are usually expressed in degrees. A full circle is divided into 360 degrees, abbreviated as 360°. Note from the figure these two particular values:
3. Each degree is divided into smaller units:
These smaller units, however, can only be measured with highprecision instruments. 
Horizontal angle BAC 

A circle has 360 degrees 
7. When you measure angles in a field, you can check on the accuracy of your measurements by applying this basic rule. Remember that the sum of the interior angles of any triangle equals (32) x 180° = 180°. 
Triangle with equal sides: 60° + 60° + 60° = 180° 

Any triangle: 65° + 75° + 40° = 180° 
Right triangle: 90° + 60° + 30° = 180° 
8. There are only a few ways to measure horizontal angles in
the field. The method you use will depend on how accurate a result you need,
and on the equipment available. Table 2 compares various methods
and will help you to select the one best suited to your needs.
Note:because 90° angles are very important in topographical surveys,
the measurement of these angles (used for laying out perpendicular lines) will
be discussed in detail.
TABLE 2
^{1}*Simple **More difficult
***Most difficult 
Homemade graphometer

Magnetic compass



Protractor Planetable 
3.1 How to use the graphometer 

1. A graphometer is a topographical instrument used to measure horizontal angles. It is made up of a circle graduated in 360° degrees. Around the centre of this circle, a sighting device can turn freely. This device, called an alidade, makes it possible to create a line of sight that starts from your eyes, passes through the centre of the graduated circle, and ends at the selected landmark or ranging pole. When in use, the graphometer is rested horizontally on a stand. 2. You can build your own graphometer by following the instructions below. It might be a good idea to ask a carpenter to help you. 
A graphometer 

Building your own graphometer 

3. Begin building your graphometer with the graduated circle given in Figure 1. You can make an enlarged copy of it, or you can draw a copy of it using tracing paper. 
Materials to build a graphometer 

4. Get a wooden board, 1 cm thick and 21 cm square. 5. Find its centre by drawing two diagonal lines on this board, from opposite corners. The point where the lines cross is the exact centre of the board. 6. Obtain one nut and bolt, about 1.5 cm long. At the centre of the board, drill a hole into which the bolt will fit tightly. On the bottom side of the board, drill the outside of the hole slightly larger to fit the nut. 
Find the centre of the board and drill a hole there 

7. Cut a hole of about the same size exactly at the centre of the graduated circle (Figure 1). Glue this sheet of paper on to the wooden board. Carefully align the central holes in the board and the circle, and make sure the four sides of the sheet are parallel to the sides of the board. You can do this easily by matching the two diagonal axes you have drawn on the board with the circle graduations 45°, 135°, 225° and 315°, respectively. 
Glue Figure 1 to the board 

8. If possible, protect the sheet of paper. To do this, get a piece of transparent plastic sheet bigger than the board, and stretch it over the front side of the board. Attach it at the back with several thumbtacks. 
Protect the paper with a sheet of plastic 
9. Now build the sighting device, called the mobile alidade. Get a thin wooden ruler, 16 cm long and 3.5 cm wide. Find its centre, as you did on the board, by drawing on it two diagonal lines from opposite corners. Draw a line through this centre point parallel to the long sides of the ruler. At the same point, drill a hole just a little larger than the diameter of the bolt. Exactly on the central line you have drawn, near each of its ends, drive a thin, headless nail 4 to 5 cm long into the ruler. Be careful not to let the nails break through to the other side of the ruler, and make sure you drive them in vertically. Your alidade is now ready. 
Find the centre of the ruler and drill a hole there 

10. To attach the alidade to the base, place a thin washer over the hole in the board you have prepared. Align the central hole of the alidade over the washer. Add a washer above the alidade and one below the wooden board, aligned with the central holes. Push the bolt through all the washers and holes and tighten the nut, so that the alidade will turn around under slight pressure. 
Bolt your alidade to the base 

11. On the board, along the 0° to 180° line, but outside the
graduated circle, drive in two headless nails like the ones placed in the
alidade. These will form a second line of sight. Clearly mark the top half
of this line of sight with an arrow pointing exactly at the 0° graduation.
12. On one end of the alidade, draw an arrow from the centrebolt, along the median line, and through the nail at the end. The tip of the arrow should point exactly to the end of the median line above this nail. This arrow will help you to determine the graduation when you need to read it. 
Mark your sighting line with nails and arrows 

13. To make your measurements more accurate, get a stake about 1.20 m high, and sharpen one end into a point. Drive this end into the ground and rest your graphometer on the other end as you measure. 
Using the graphometer with a stake to steady it 
Using the homemade graphometer to measure horizontal angles 

14. Orient the graphometer with its 0° to
180° sighting line on the left side AB of the angle you need to measure.
Position the graphometer so that its centre, the bolt, is exactly
above point A on the ground, the station, from which you will
measure horizontal angle BAC. For more accuracy, you can use
a plumbline (see Section 4.8). If you have attached your
graphometer by its centre to a stake, drive the pointed end of the stake
vertically into the ground at the angle's summit A.
15. Check that the graphometer is as horizontal as possible. To do this, place a round pencil on the board. If the pencil does not roll off, turn it 90° degrees and check again. When the pencil does not roll off in either direction, the graphometer is horizontal. 
Angle BAC 

When the pencil does not roll off in either direction, the graphometer is horizontal 
16. Check again that the 0° to 180° sighting line is aligned well with the left side AB of the angle you need to measure. Adjust it if necessary, making sure not to disturb either the position of the station or the horizontality of the graphometer.
Sight from A to B 
Make sure the 0°180° line is on line AB 
17. Move the mobile alidade around to the right until its sighting line lines up with the right side AC of the angle BAC.
18. Read the graduation above the arrow on the central line of the mobile alidade. This is the value of angle BAC in degrees.
Now sight from A to C, moving the alidade but not the graphometer 
Note: it is easier to position the graphometer above the station on the ground and make it horizontal without sighting with the 0° to 180º line. Just make sure that the left side AB of the angle is to the right of this 0° to 180° line. Then take two readings, using the mobile alidade for both the left side AB and the right side AC of the angle. The value of the angle equals the difference between these two readings.
Example

Set the graphometer at point A 

Take the readings of lines AB and AC 

Measuring an angle with an inaccessible summit 

19. To use the preceding method, you must be able to reach the summit A of the angle. When you cannot reach the summit, you have a choice of two methods. 
Measure angles CBA and BCA to calculate angle BAC 

20. You can set out a line CB from any point on
one of the angle's sides to any point on its other side, making a triangle
within the angle. Measure the two angles made by this new line and the angle's
sides. The angle at the inaccessible summit of the triangle you have made
equals the difference between 180° and the sum of the other two angles.

Summit A is across a river 

Example You cannot reach summit A to measure angle XAY. From point B on line AX, set out Iine BC, where point C is on line AY. At station B, measure angle CBA = 60°; at station C, measure angle BCA = 73°. Calculate angle XAY = 180°  (60° + 73°) = 47°.

Angle BAC = angle XAY 
21. Or you can lay out two perpendicular lines (see Section 3.6) from two points on one of the angle's sides. On each of these new lines, measure an equal distance. Join these two points with a line, which will be parallel to one of the angle's sides. Prolong the line until it intersects the other side of the original angle. At the intersection point, measure the new angle which is equal to the summit angle. 
Summit A lies behind an obstacle 

Example You cannot reach summit A to measure angle XAY. On line AX, mark two points B and C. From these points, lay out perpendiculars BZ and CW. On these perpendiculars, measure segments of equal length from line AX, calling them segments BD and CE. Connect points E and D to form a line, which is parallel to AX. Then extend line ED until it intersects line AY at point F. From a station at point F, measure angle EFY. Its measurement will be the same as that of angle XAY. 
Set out perpendiculars BZ and CW 

Find line ED, and prolong it to F 

Measure angle EFY 
Angle EFY = angle XAY 
Measuring consecutive angles 

22. At one station, you may have to measure several angles formed by a series of lines which meet at one point, called converging lines. The angles they form are called consecutive angles. 23. To measure consecutive angles from one station, first measure all the angles by using the line furthest left as the reference line. Then, by simple substraction, you can calculate the individual angles. 
XPA, APB and BPC are consecutive angles 
Example
At station P, you have to measure three consecutive angles, XPA, APB and
BPC. Take PX (furthest to the left) as the reference line and align the 0° graduation
of the graphometer with it. Keeping the graphometer fixed in that position,
move the mobile alidade around and measure each angle in turn (in this case,
angles XPA = 40°, XPB = 70° and XPC = 85°). Calculate the consecutive angles
as follows: XPA = 40°, as directly measured;
APB =XPB XPA = 70°  40° = 30°; BPC = XPC  XPB = 85°  70°= 15°.
Measure each angle from the 0° line... 
... then calculate the individual values 
3.2 How to use the magnetic compass 

What is a magnetic compass? 

1. A simple magnetic compass is usually a magnetic needle which swings freely on a pivot at the centre of a graduated ring. The magnetic needle orients itself towards the magnetic north*. The needle is enclosed in a case with a transparent cover to protect it. 2. Orientation compasses are often mounted on a small rectangular piece of hard transparent plastic. They have a sighting line in the middle of a movable mirror. If you tilt this mirror, you can see both the compass and the ground line. 
A simple compass 

An orientation compass 
A prismatic compass 

3. Prismatic compasses give more accurate readings. To use one, hold it in front of your eyes so that you can read its scale. You can see the scale through a lens by means of a prism. Then turn the compass horizontally until the crosshair is aligned with the ground mark (an optical illusion makes the hairline appear to continue above the instrument's frame). At the same time, the reading is shown on the compassgraduated circle behind the actual hairline. Since the graduated ring automatically orients itself, the reading directly gives the measure of the angle between magnetic north* and the line of sight (see also next paragraphs).  
4. A magnetic needle always points in the same
direction  the magnetic north. This is why compasses are often
used for orientation in the field and for mapping surveys (see, for example,
Section 7.1 of this manual). The part of the compass needle pointing to
the magnetic north is clearly marked, usually in red or a dark colour.
5. The outside ring of a compass is usually graduated in 360°. The 0° or 360° graduation is marked N, which means north. In most compasses, the graduation increases clockwise and the following letters can be read on the circle:
Intermediate orientations, such as NE, SE, SW and NW, are also shown sometimes. 
Using the compass to measure horizontal angles 

6. You have learned that the needle of a compass always points in the same direction  the magnetic north. To use this direction as a reference, you need to make sure that the 0° graduation lines up with it. If the 0° graduation of your compass does not line up exactly with the magnetic north, turn the external ring until it does. Only then can you use your compass as described below.  
7. At any station, the angle formed by the magnetic north and a straight line is called the azimuth of that line. This magnetic azimuth from north, called azimuth or Az, is always measured clockwise, from the magnetic north to the line. Example Azimuth OA = 37°;

Measuring the azimuth of a line 

8. To measure the azimuth of a line, take a position at any point on the line. Holding your compass horizontally, sight at another point on the line, such as a ranging pole. To do this, align the sighting marks of the compass with this point. If necessary (as with some orientation compasses), first align the 0° graduation N exactly with the northern point of the magnetic needle. At the intersection of the sighting line and the graduated ring, read the azimuth of the line from the point of observation. The reading will be most accurate if you limit the length of the sighting line to 40 to 120 m. Place more ranging poles on the line as you need them. Note: to check the value of an azimuth, turn around and look in the opposite direction at another point on the same line. Read the measurement of this azimuth, which should differ by 180°. Usually, the difference will not be exactly 180°. If the difference is small enough, you can ignore it, or correct it by averaging the two readings. If it is large, you have made an error which you should correct. 
Check the azimuth in both directions 

Example
To determine the azimuth of line XY, marked by ranging poles B and C. stand with the compass at station A in the middle of the line. Sight forward with the compass at ranging pole B and read azimuth AB = 65°. This is the forward azimuth. Check this value by turning around, sight backward with the compass at ranging pole C and read the rear azimuth, AC = 245°. The difference between the two azimuths is 245°  65° = 180°, which means that the measurements are accurate. 

Forward azimuth = 65° 
Rear azimuth = 245° 
Measuring a horizontal angle 

9. To measure a horizontal angle, stand at the angle's summit and measure the azimuth of each of its sides; calculate the value of the angle as follows. 10. If the magnetic north falls outside the angle, calculate the value of the angle between the two lines of sight as equal to the difference between their azimuths. Always subtract the smaller number from the larger one, no matter which azimuth you read first. Just be sure that the magnetic north is not inside the angle. Example (a) Angle BAC; Az AB = 25°; Az AC = 64°;


(b) You have to measure angle XAY; measure the azimuth of AX = 265°; measure the azimuth of AY = 302°; the angle XAY measures 302°  265° = 37°.  
11. If the magnetic north falls inside the angle, the angle
between the two lines of sight is equal to 360° minus the difference
between their azimuths. To calculate the angle, first find the difference
as you did in step 10, above, then subtract this number from 360°.
Example You have to measure angle EAF; measure the azimuth of AE = 23°; measure the azimuth of AF = 310°; angle EAF measures 360°  (310° 23°) = 73°.
Note: to check on your measurements and to improve their accuracy, you should repeat each measurement three times from the same station. These measurements should give similar results. 

12. If you cannot reach the summit of the angle, separately measure the azimuth of each line from another point situated on it (see step 8, above) and calculate the angle as in step 9, above. Example You have to measure angle BAC, but the summit A is not accessible;
at point X on AB, measure azimuth XB = 39°; at point Y on AC, measure
azimuth

Surveying a polygonal site 

13. When you must survey a polygonal* site, measure the azimuth of two sides from each of the summits. For each side of the polygon, you will thus determine one forward and one rear azimuth. You can then check on the accuracy of the two azimuths, which should differ by 180°. If they do not, subtract 180° from the greatest azimuth and calculate the average between this value and the smallest azimuth. To do this, add the two numbers and divide by two. From averages like this for the other pairs of azimuths, you can calculate the interior angles of the polygon, as explained above. Note: to make a final check, add all the interior angles. This sum should equal (N  2) x 180°, N being the number of sides of the polygon (see Section 3.0, step 6). 

Example
You have to survey polygon ABCDEA. From summit A measure forward Az AB = 40° and rear Az AE = 120°. Move clockwise to summit B and measure rear Az BA = 222° and forward Az BC = 110°. Proceed in the same way from the other three summits C, D and E. In total, you get ten measurements. Mark them down in your notebook. (See columns 1 and 2 where the order of measurements is shown in parentheses.) Calculate the values of column (3) by subtracting 180° from the largest azimuth measured at each summit. This gives you values which should be equal to the smallest observed azimuths, written either in column 1 or in column 2, according to the position of the summit. 



The observed forward and rear azimuths of polygon ABCDE, in a table and in a sketch 

Calculate interior angles, combining the calculated azimuths (columns 4 and 5) two by two as follows, with the help of a little sketch:









Check your calculations: the sum
of the angles should be equal to (5 2) x 180º = 540º.
These calculations (79º + 112º + 104º + 118º + 127º = 540º) are correct.
14. If you have to measure consecutive angles, proceed as described earlier (see end of Section 3.1).
When using a magnetic compass to measure horizontal angles, you should carefully check the following points:
15. The magnetic needle must swing freely on its pivot. Keep the compass horizontal in one hand and, with the other hand, bring an iron object close to the magnetized needle's point. Make the needle move to the left with the iron; when you move the iron away, the needle should swing quickly and smoothly to its original position. Repeat the movement in the opposite direction to double check
Put the iron near the compass to attract the needle

... then pull it away. The needle should swing
back into place


16. The magnetic needle must be horizontal when the compass is horizontal. Lay the compass on a horizontal wooden surface (such as a table) and check that the needle remains horizontal. If it does not, you will have to open the case of the compass and add a light weight to the needle. To do this, you can wind some cotton sewing thread around the part of the needle that is highest, and move the thread back and forth until the needle is balanced and horizontal. 
If the needle is not horizontal


... wrap some thread around it for balance

17. Do not keep iron objects close to the compass. Iron will attract the magnetic needle, and your measurements will be wrong. Distance measuring lines made of metal, such as steel bands, steel tapes and chains, as well as metal ranging poles and marking pins, should be kept 4 to 5 m away from the compass when you are measuring angles. If you wear eyeglasses with metal frames, you will also have to keep them away from the compass. Remember that concrete structures (towers, bridges, etc.) are built with iron bars which may also cause the compass needle to move.
18. Do not use a compass when there is thunder. It affects the needle.
19. Do not use a compass near an electric power line.
20. Keep the compass horizontal while you are measuring with it.
Metal and electricity can affect your compass


Note: because the magnetic needle of the compass is always affected by the presence of iron nearby, checking the measured azimuths (as explained earlier) is extremely important. If your results do not agree after repeated measurements, local magnetic disturbances caused by the presence of iron in the ground may be responsible for the errors. You should then use another method of measurement.
To use the graphic methods for measuring horizontal angles, you need to draw the angle on paper first. Then you will measure the angle with a protractor (see step 11, below). As you have seen with other methods, you can obtain more accurate results if you repeat the procedure at least twice to discover possible errors.
1. With this method, you can use a simple magnetic compass (see Section 3.2.). The only purpose of this compass is to show the direction of the magnetic north*.
2. Get a 30 x 30 cm piece of stiff cardboard or thin wooden
board, and several sheets of squareruled paper (such as millimetric
paper). Lightly glue each sheet, at its four corners, to the board, one
on top of the other.
3. On the upper lefthand corner of the top sheet, attach the compass, for example with a string or rubber band or within a small wooden frame, so that its 0º to 180º reference line is parallel to one of the rules on the paper. With a pencil, draw an arrow straight up toward the top of the sheet, and mark it North. 
4. To draw the horizontal angle BAC you need to measure, stand at the angle's summit A and look at aground line AB which forms one of the sides of the angle.
5. Keeping your board horizontal on the palm of one hand held in front of you, turn it slowly around until the northern point of the compass needle reaches the 0°graduation Your sheet of paper is now oriented, with its arrow facing north.
Note: it will be easier if you rest the board on a stable support, such as a wooden pole driven into the ground.
Line up your board so the compass points
to magnetic north 
Rest the board on a support for stability

6. Without moving the board any more, trace on the paper in pencil, with your free hand, a line ab going straight ahead in the direction of the ground line AB.
7. Repeat the procedure described in steps 5 and 6 above, looking at the ground line AC which forms the other side of the angle and drawing line ac.
Sight and draw line ab

Then sight and draw line ac

8. Using a protractor (see steps 15 to 17 below), measure the azimuths of the lines you have traced as the angles formed by them with any of the paper rules running parallel to the north. Remember to measure the angle clockwise from the north to the pencil line (see Section 3.2).
Note: you only need to measure angles smaller than 90º, since the squareruled paper shows the 90°, 180º and 270º directions.
9. Take the azimuths of the two sides of the horizontal angle, and calculate
the value of the angle as described in Section 3.2.
10. If you have a planetable (see Section 7.5), you can use it to draw the angles on paper while you are in the field. Then it is easy to measure them with a protractor (see steps 15 to 17, below).
Measure the azimuths with a protractor

11. A protractor is a small instrument used in drawing. It is graduated in degrees or fractions of degrees. The semicircular protractor is the most common type, but a fullcircle protractor may be best for measuring angles greater than 180º. Protractors are usually made of plastic or even paper. You can buy one cheaply in stores that sell school supplies. Or you can use the one provided in Figure 2. Either make a copy of it, or copy it on to transparent tracing paper. Note that the arrow points at the exact position of the protractor's centre.
Making your own protractor 

12. Cut the drawing of the protractor in Figure 2, exactly along its curved outer edge.  13. Glue this paper protractor onto a slightly larger piece of stiff cardboard.  
14. Cut the cardboard, exactly following the shape of the protractor. 
Using the protractor to measure an angle you have drawn 

15. AIign the 0º to 180º straight edge of the protractor with one of the angle's sides AB.  16. Move the protractor so that its centre is positioned exactly on the summit A of the angle, keeping the 0º to 180º straight edge on the angle's side AB.  
Line up the protractor

Centre it at the angle's summit


17. Look for the point where the angle's other side AC intersects the graduation on the round edge of the protractor. Read the value of the angle from this graduation. This value may be expressed in degrees and minutes (remember that half of 1 degree equals 30 minutes).  Note: if the sides of the angle are not long enough to intersect the edge of the protractor, lengthen them before you begin measuring.  
Read the graduation

Prolong lines if necessary

3.4 How to measure horizontal angles by the rightangle method 

1. The rightangle method is best for measuring angles smaller than 10 degrees in the field, since the preceding methods do not give accurate results. The rightangle method is based on the geometrical properties of right angled triangles (see Section 3.0, step 7).  2. From the angle's summit A, measure 10 m along one of the sides AC of the angle. Clearly mark this point D, with a ranging pole for example.  

Measure 10m


3. From point D, lay out a perpendicular line and prolong it until it intersects the second side of the angle. Clearly mark this intersection point E.  4. Accurately measure the length, in metres, of this perpendicular line DE.  
Lay out a perpendicular

Calculate the value

5. Divide this length by 10 to obtain the tangent* of the angle.
6. Look for this value in Table 3 and find the measurement of the angle BAC in degrees and minutes.
Example
You have to measure the small angle XAY:

Measure 10 m to point B





Lay out perpendicular BZ

From distance BC = 1.12 m, calculate the value
XAY = 6º20'

TABLE 3

3.5 How to measure horizontal angles with a theodolite 

What is a theodolite? 

1. A theodolite, sometimes called a transit, is an expensive instrument which surveyors use to measure horizontal angles precisely. It is like a graphometer, but more complicated (see Section 3.1). Most theodolites are designed to measure vertical angles as well. The theodolite's basic features for measuring horizontal angles are:

2. If you want to measure angle BAC, place the theodolite horizontally on its tripod at summit A. Set the index on the horizontal graduated circle at zero and take a sighting to B. Clamp the circle in position. Turn the telescope and its circular plate to take a sighting to C, while rotating through the angle BAC. You may read the angle measurement directly from the circular plate index.
Sight at the points...

... and read the measurements

3.6 How to set out right angles or perpendiculars 

What are right angles and perpendiculars? 

1. A right angle is a 90º angle. Two lines intersecting each other at a right angle are called perpendiculars. You have already learned how right angles can be useful for measuring distances (see Section 2.9) and for measuring horizontal angles (see Section 3.1, step 2.1). Right angles are also often used in fish culture, for example when you build rectangular ponds, to estimate the volume of a future reservoir (Vol.4, Water, Section 4.2), or measure land areas (see Chapter 10 of this manual). 
DC is perpendicular to AB


Some uses of perpendiculars

2. There are two main problems to be solved:
Laying out a perpendicular from point X on line
AB

Dropping a perpendicular by the fullcircle method 

3. Set out line XY with ranging poles, and point A above or below it with a marking pin. You will drop the perpendicular from point A to XY. Get a simple line (a liana or rope) or a measuring line (a tape or chain) slightly longer than the distance from point A to line XY. 
Attach the line to the pin


4. Attach one end of the line to the marking pin at point A, keeping it near the ground. 
Put a marking pin at point A


5. Walk with the other end of your measuring line to the line XY, and stop about 2 m beyond the point where you crossed it.  
6. Holding the line in your hand, trace an arc with it on the ground. To do this, swing the line in a curve to the left until you intersect XY, and mark this point B. Then swing the line in a curve to the right until you intersect XY, and mark this point C. 
Trace an arc on the ground and mark
the points of Intersection 
7. Measure on XY the distance BC between these two marked points. 8. Divide this distance by 2 and measure this new distance from point B. Mark this point D  it will be exactly in the middle of BC. 
Measure distance BC


9. Connect point D and the original point A to form a new line AD perpendicular to XY. 
Divide by 2 to find midpoint D


Connect points D and A to form the perpendicular

Set out line XY and point A as described above. Prepare a measuring line a little longer than half the distance from point A to the line XY on which you will drop the perpendicular.
10. From any point B on line XY, measure distance AB to point A.
11. Divide this distance AB by 2 and mark the centre point C.
12. Attach one end of your measuring line to point C as in step 4, above.
Mark point A

Measure distance AB


Divide by 2 to find midpoint C

13. Walk with the other end of your line to point B on XY, and clearly mark this distance CB on the measuring line.
14. Trace an arc on the ground with the line length CB. To do this, swing the line in a curve to the right until you intersect XY again. Mark this point D.
15. Join D to the original point A to form a new line AD perpendicular to XY.
Find length CB on your rope

Trace an arc on the ground to find point D


Connect point A and D to form the perpendicular

Setting out a perpendicular by the midpoint method 

16. The easiest way of setting out a perpendicular from a fixed point A on line XY is to use a simple line clearly marked at its midpoint with, for example, a knot. This line can be a liana, a rope or a string or you may use a measuring tape, whose graduations will help you to locate the exact midpoint. For good results, the line should be at least 8 m long. A longer line will make your measurements even more accurate. If you are working alone, make a small loop at each end of the line.  
17. Set out line XY and mark point A from which you will set out the perpendicular. On either side of point A and along line XY, measure the equal distances AB = AC of about 2 m. You can use part of the measuring line to do this. Mark points B and C with stakes.
18. Loop one end of the line over stake B and the other end over stake C.
Mark point A

Measure 2 m each way to find points B and C
Loop the ends of the line over stakes B and
C 

19. Taking the line by its midpoint D, stretch the line tightly, making sure that the two ends are still looped over stakes B and C. Mark the position of the midpoint D with a stake. Line DA will be perpendicular to line XY. 
Stretch the line tight from its midpoint to find
point D


Connect points A and D to form the perpendicular

20. To set out a perpendicular by the intersection method, you can use a simple line again. The method you use will depend on the length of the line. Remember that:
Using the shortline intersection method 

21. To use this method, you will need a simple measuring line such as a liana or a rope 5 to 6 m long, a short pointed stick or thin piece of metal (such as a big nail) and five marking stakes. 22. Set out line XY. On this line, choose point A, from which you will set out the perpendicular, and mark it clearly with a stake. 23. With part of your measuring line, measure a 2 to 3 m distance to the left of point A on XY. Mark this point B with a stake. 24. Measure the same distance on XY to the right of point A. Mark this point C with a stake. 
Mark point A


With the rope find point B


... and, by making an arc, point C


25. Make a fixed loop at one end of your line,
and securely attach the pointed stick or piece
of metal to the other end.
26. Place this loop around marking stake B. Then, keeping the line wellstretched, trace a large arc on the ground with the other end of the line. This arc should extend beyond point A, and a long way on either side of XY. 27. Take up the loop from stake B and place it around stake C. Trace another arc on the ground which should intersect the first arc at two points, D and E. 28. Clearly mark these two points D and E with stakes. 

Do the same from point C  the arcs intersect
at points D and E 
From point B, trace a large arc on both sides of
line XY


29. Taking up the loop from stake C, place it around stake D; holding the other end of the line, walk to stake E and attach it there; check to see if the line touches the central stake A (remember that the perpendicular was originally to be set out from point A); if it does, line DE forms this perpendicular on the ground. 
Attach the line at point D and walk to point E


The stretched line forms perpendicular DAE

Using the longline intersection method 

30. To use this method, you will need a simple line about 55 m long, a short pointed stick or thin piece of metal and four marking stakes. 31. Clearly mark point A on line XY with a stake. You will set out the perpendicular from this point. 32. Measure 25 to 30 m to the left of point A on line XY, using part of your measuring line; mark this point B with a stake. 33. Measure the same distance on XY to the right of A; mark this point C with a stake. 34. Make a fixed loop at one end of your line. Securely attach the pointed stick or piece of metal to the other end of the line (as in step 25, above). 
Mark point A


Use the line to find point B...


... and point C

35. Place the loop around marking stake B and, with the other end of the line in one hand, walk diagonally away from line XY. When you reach a point above A where the line is well stretched, trace an arc 2 to 3 m long on the ground with the end of your line. 36. Repeat the last step from the second stake C. The arc you mark on the ground from this point should intersect the first arc at point D. 37. At this intersection point D, drive a marking stake into the ground. 38. The line AD joining D with the original point A is perpendicular to XY. Note: you can only use the intersection method on ground that is clear of large rocks and high vegetation, because you must be able to mark and see the arcs easily. If necessary, you can clear the ground as you work. 
From point B, trace an arc above point A


Do the same from point C  the arcs intersect
at point D


Connect points D and A to form the perpendicular

39. The 3:4:5 rule is that any triangle with sides in the proportion 3:4:5 has a right angle opposite the longest side. The method is based on this rule.
The length of the simple line you use for measuring will depend on the length of the perpendicular you are setting out. The longer the perpendicular, the longer your measuring line must be.
Examples
40. To make your simple line, get a rope 11.5 cm thick; it is best if made from natural fibres which will stretch or shrink very little. A piece of used sisal rope will stretch or shrink less than a new one. You can also use a measuring tape.
41. There are several ways of using the 3:4:5 rule method, depending on the type of measuring line you use and the number of people who can work together with you. When using medium or long lines, it is best to work in a team of three people. When using a short or very short line, you can work by yourself.
Making your own 3:4:5 measuring line 

42. You can easily make a simple line to use with the 3:4:5 rule method. This line is sometimes known as a ratio rope. The following shows one way of making a short line about 13 m long, but you can make shorter or longer lines in the same way. 43. Take a piece of rope about 13 m long. A few centimetres from one end, tie a metal ring to it securely with heavy string. 


44. Measure a length of 3 m along the rope from this ring, and attach a second ring to the rope. 45. Using a measuring tape, check that the distance from the first to the second ring is exactly 3 m. If it is not, adjust it. 
Tie the ring on securely


Measure again when the ring is in place

Attach the second ring 3 m down

46. Measure a length of 4 m from the second ring and attach a third ring. With a measuring tape, check that the distance is exactly 4 m. Adjust it if necessary. 47. Measure a length of 5 m from the third ring. Attach this end point of the rope to the first ring. Check the length with a measuring tape and adjust if necessary. 
Attach the third ring 4 m down and
check the measurement 

5 m from the third ring...


Check the final measurements

... attach the rope to the first ring

Using the short 3:4:5 line to set out a right angle 

48. Set out straight line XY on which you want to construct the right
angle using a short line. Get several wooden or metal stakes 
Stake the first ring at point A


50. Stretch the 3 m segment tightly along line XY, and pin or stake it at point B, using the ring between the 3 m and 5 m segments.  
Stretch the 3 m segment to point B...

... and stake it down

51. Hold the ring between the 4 m and 5 m segments, and pull
the ratio rope into the shape of a triangle, making sure that the 4 m and
5 m segments are tightly stretched. Using this ring, stake the
rope at C.
52. The angle which has been formed at A between the 3 m and 4 m segments of the ratio rope is a right angle. 
With the third ring, walk away from line XY


When the rope is stretched tight, mark point C
to form the right angle 
Note: you can also use a 3:4:5 ratio rope with much shorter segments. A 30 cm: 40 cm: 50 cm line, for example, is ideal for measuring angles of smaller areas, which can be used for laying out a Vknotch weir, for example (see Vol.4, Water, Section 3.6).
Using the medium 3:4:5 line to set out a right angle 

53. Use a line about 36 m long, prepared like the short line, except that the sections should be 9 m, 12 m and 15 m long. Starting at point A, where the right angle has to be set out, stretch the 12 m segment along line XY; at this point attach the ring on the line to stake B. 54. With the 15 m segment, walk away from B while your assistant returns to the original point A with the 9 m segment of the line. 55. When these last two sides of the triangle are fully stretched, mark the point C which connects the 9 m and 15 m lengths. This point forms perpendicular AC at A. 

With a longer line, use an assistant

Using the long 3:4:5 line to set out a right angle 

56. On a rope about 65 m long, clearly mark the 0 m, 15 m, 35 m and 60 m lengths. You should work in a team of three people when using this line. 57. The first person holds the rope at the 15 m mark at point B on line XY, from which the perpendicular is to be set out. 58. The second person holds the 0 m and 60 m marks together at point A along XY. 

Keep the 0 m and 60 m marks together at point A


At point B, the second person holds the 15 m mark

59. The third person takes the rope
at the 35 m mark and walks away from XY. He or she adjusts his or her position
until the two sides of the triangle are stretched. When this is done, the
position point C is marked. Joined to point B, this forms perpendicular
BC to XY.
Note: you should always doublecheck distances to make sure that no errors have been made. 
The third person walks away, holding the 35 m mark


When the rope is stretched tight, mark point C
to make the right angle 
You need to set out the centreline WZ of a dike perpendicular to the centreline XY of another dike, for example. Using a measuring tape at least 80 m long and working in a team of three, proceed as follows:
60. From the intersection point A of the two dike centrelines, measure 40 m along XY, the known centreline. Mark this point B.
Measure 40 m to point B


61. While one person holds the 0 m graduation of
the tape at point B, the second person holds the 80 m graduation
at point A, where the two centrelines intersect.
62. The third person, holding the tape at the 50 m graduation, walks away from XY until the tape is fully stretched. He or she clearly marks the place where he or she stands point C. This point defines the second centreline WZ perpendicular to the first one. 
Walk away from A and B holding the 50 m mark


Setting out a perpendicular with a crossstaff 

63. A crossstaff is an inexpensive sighting instrument which is very useful for setting out right angles. There are several models, such as the octagonal brass crossstaff, which has sighting slits cut at right angles to each other, and the foresight/backsight model. In use, crossstaffs must be firmly fastened to a support, usually a stake driven vertically into the ground. Their useful range does not extend beyond 30 to 40 m. You might be able to borrow a crossstaff from a surveyor's office, or you can build your own as described below. Note: the octagonal crossstaff also has additional sighting slits cut at 45°, and is useful for setting out 45degree angles (see, for example, Section 2.9, step 7). 
Professional crossstaffs


Sighting with a homemade crossstaff

Making your own crossstaff 

64. Get two metal or wooden strips 2 to 3 cm wide and 20 to 25 cm long. Find the centres of the strips at the intersection of two diagonal lines, as you found the centre of your alidade in step 9. Drill a small hole exactly at this centre point you have found on each strip. These are the crosspieces.  
65. Provide a sighting line on each crosspiece. To do this on wooden strips, drive in one small headless nail, centred, near each end of each strip. On metal strips, you can weld or glue small nails or metal points near the ends of the strips  
66. Place the crosspieces approximately at right angles and, with a screw, attach them loosely in that position to the top of a 1.50 m vertical stake. If you use washers between the wooden crosspieces and the stake, it will be easier to tighten the strips securely later. 
Adjusting the homemade crossstaff 

67. Lay out a right angle on the ground using a long 3:4:5 ratio rope (see steps 56 to 59, this section). The sides of the triangle will be next 15 m, 20 m and 25 m long. 68. Put a short stake at point A, the corner of the right angle, between the 15 m and 20 m sides. Then put ranging poles at points B and C to mark the sides of the angle. 69. Position the crossstaff and its vertical support at point A. 70. Align one crosspiece alongside AB and sight towards point E. 71. Without moving the vertical support, align the second crosspiece along the other side AC of the angle, and sight towards point C. Tighten the screws a little to keep the crosspieces in place. 
Lay out a right triangle


Sight point B with one crosspiece

Sight at point C with the other one
and tighten the screw 
72. Rotate the vertical support 90º to check that the crosspieces are truly at a right angle. Sight at points B and C again and correct the position of the crosspieces if necessary.
73. Repeat this process until you are sure that each crosspiece is aligned with one side of the right angle, that is, that they are at 90° angles from each other.
74. When both crosspieces are properly aligned, firmly tighten the screw holding them to the vertical support.
75. Check both sighting lines again after tightening the screw to make sure that the crosspieces have not slipped.
76. To help you adjust the crosspieces later, cut or engrave (with a large nail) marks in the wood or metal of the bottom crosspiece when the top piece is in position.
Rotate the crossstaff to check its accuracy


Mark the lower crosspiece for reference

77. To use the crossstaff, you will need an assistant.
78. Lay out straight line XY on which you need to construct the right angle at point A.
79. Place the support of the crossstaff in a vertical position at point A.
80. Ask your assistant to hold a ranging pole in a vertical position at point B, near the end of XY.
81. Sight along one of the crosspieces and rotate the vertical support until the sighting line is aligned on B.
82. Without moving the crossstaff and its vertical support, sight along the other crosspiece. At the same time, direct your assistant to stand with a ranging pole as near to this sighting line as possible.
Sight along XY to align the crossstaff

Sight along the other crosspiece to
find the perpendicular 

83. Tell your assistant to hold the ranging pole in front of him or her and move to the right or left until the pole is exactly on the sighting line AZ.
84. When you are sure he or she is on line AZ, tell him or her to mark his position with stake C.
85. The angle BAC you have determined at point A, where the crossstaff is placed, is a 90° angle.
Place the ranging pole at point C on line AZ

BAC is a right angle

Note: with the help of a crossstaff you can easily determine the rectangular areas which you need for a fish pond lay out. You can also build a grid of squares by determining intermediate angles along your straight lines. This is a method used in estimating reservoir volumes, for example (Volume 4, Water, Section 4.2).
1. Parallel lines, also called parallels, are lines equally distant from each other at every point. Parallel lines run side by side and will never cross. They are very important in fish culture and are often used in designing fish farms (for example, for parallel dikes and ponds), in building dams and in setting out water canals. Parallels are also useful when laying out lines under difficult conditions (see Section 1.7).
Setting out parallels by the 3:4:5 rule 

One way to set out a parallel line uses the 3:4:5 rule. It works like this: 2. On given line XY, select two points A and B which are fairly distant from each other (for example, 20 to 30 m apart), and mark them with pegs. 3. From each of these points, set out a perpendicular using the 3:4:5 rule method. Remember that the length of the line you will use depends on the length of the perpendicular you are setting out (see Section 3.6, step 35). 4. Prolong these two perpendiculars as required. Then, measure an equal distance from the given line XY on each of them; mark these two points C and D. 5. Through these two points, set out a line WZ. This line will be parallel to XY. 
Set out two perpendiculars...


Prolong each one equally

Connect the points to form the parallel.

Setting out parallels with the crossinglines method 

To use the crossinglines method, you do not need to set out perpendiculars; you will only measure distances. However, this method cannot be used when you need to measure the exact position of the parallel you need to set out. It is useful when the distance of the parallel is not important, such as when you are prolonging a line over an obstacle (see Section 1.7). Proceed as follows: 6. Lay out line XY. Select any point A which will belong to the parallel line you need to set out. Clearly mark point A with a peg. 
Choose point A


7. From point A, set out an oblique line AZ. Mark point B where AZ intersects the original line XY. Note: an oblique line is neither parallel, nor perpendicular. 8. Measure the length of the oblique line AB. 9. Divide this length by 2, measure this distance from point A, and mark this central point C. 
Divide AB by 2 to find point C


Choose point D

Lay out AZ, find point B, measure distance AB

10. On the original line XY, select point D, which should lie as nearly opposite point A as possible. 11. From point D, set out a straight line DW passing through point C. 
Set out line DW


12. Measure distance DC. 13. From point C on line DW, measure a distance equal to distance DC. Mark the end of the segment point E. 14. Connect points E and A with a line KL. This line is parallel to line XY. 
Find point E


Connect the points A and E to form the parallel

15. When you build a fishfarm, you will often need to set out a series of rectangular plots on the ground. These plots are the future sites of ponds or other constructions, (see the next manual in this series, Constructions for Freshwater Fish Culture).
16. First select the direction of the main dike's centreline XY, and set it out on the ground with ranging poles. Using the measurements along this line, you will be able to mark the points A, B and C where you will set out the centrelines of the secondary dikes. To do this, proceed as follows: 
Set out the centre line of the main dike and the
points
where the other dikes will cross it 

17. Set out a few perpendiculars* on XY, using one of the methods given in Section 3.6, for example, from two extreme points A and B (near the endpoints of XY) and from one intermediate point C. 
Lay out perpendiculars from these points


18. Starting from points A and B, measure equal distances AF and BG along their perpendiculars; these distances should be equal to the selected distance between the main dike centreline XY and the centreline of the opposite dikes. Mark the two points on the perpendiculars, F and G, with ranging poles. 
Measure equal distances along the perpendiculars

19. Clearly set out line WZ through points F and G using
ranging poles.
20. Starting from point B on line XY, measure intermediate distances BE, EC and CD. Then move back to line WZ; starting from point G, measure intermediate distances GH, HI and IJ equal to BE, EC and CD, respectively. Mark points H, I and J with pegs. 21. While you are doing this, check that point I falls exactly on the intermediate perpendicular set out from C. If there is a small difference, adjust the positions of the perpendicular and point I. If there is a large difference, check your previous work for errors. 22. As a final check, be sure that the last measurement JF lines up with point F. 
Set out parallel WZ


Set out and check intermediate distances
