Chain Survey

An offset is a lateral measurement taken from a chain line to a point of detail. There are two types.

1. Perpendicular offsets

Measured at right angles to the chain line.

• Setting the right angle:
  • With a cross-staff: the surveyor stands on the line, sights along the line through one slit, and directs the assistant to the object along the perpendicular slit.
  • With an optical square (prism square): two mirrors at $45^\circ$ show the line and the object in one image when the surveyor is at the foot of the perpendicular.
  • With a tape (3-4-5 method): a triangle with sides 3, 4 and 5 m (or multiples) gives a right angle.
  • For short offsets, swing the tape: hold the zero at the object and swing the tape across the line; the shortest reading on the chain line is the foot of the perpendicular.
• Taken for details close to the line, usually within 15 m.

2. Oblique offsets (tie lines)

Measured at any angle, when the object is far, or a perpendicular is difficult.

• Two-distance method: the object $P$ is joined to two points $A$ and $B$ on the chain line (with $P$ in between), and the distances $AP$ and $BP$ are measured. $P$ is plotted by arcs.
• Angle and distance: the angle at a point on the chain line and the length to the object are measured.
• Used for buildings, corners of fields, when the object is inaccessible.

 chain line  A---+----------+---B
                 \         /
               AP \      / BP     oblique offsets
                   \   /
                     P

Booking

Offsets are booked in the field book at the chainage where they are taken, e.g. chainage 45 with offset 8 on the right and chainage 60 with oblique offsets 12 and 14.

The chainage runs from the bottom of the page upwards, and the offset is written beside the chain line on the side where it lies, with the object name or a sketch.

The field procedure of chain surveying follows these steps.

1. Reconnaissance

The surveyor walks over the area, observes its shape, details and obstacles, and plans the survey lines.

2. Selection and marking of stations

• Select main stations so that the main lines form well-conditioned triangles (angles $30^\circ$ to $120^\circ$).
• Select the base line (the longest, on level ground), the check lines and tie lines.
• Mark stations with pegs or ranging rods, and fix them to permanent objects (for recovery).

3. Index (key) sketch

A neat freehand sketch (not to scale) shows the whole area, the stations, the lines and the main details. Station names (A, B, C ...) are written on it.

        C
       /|\ 
      / | \ 
     /  |D \     A, B, C = stations
    A---+---B    D = check line CD

4. Ranging and chaining the lines

• Each line is ranged between its end stations with ranging rods so that the chain stays in a straight line.
• The leader and follower pull the chain; the follower stands at the starting station, and the leader carries 10 arrows, sticking an arrow at each chain length.
• The follower collects the arrows; the tally of arrows and the tape count gives the length. Main stations are then chained.

5. Taking offsets

• Perpendicular offsets are set with a cross-staff, optical square, or the 3-4-5 tape method and measured with the tape.
• Oblique offsets or ties are used for far or hidden objects.

6. Booking

All chainages and offsets are written in the field book, from the bottom of the page upwards, with objects sketched. The left and right offsets are placed on either side of the chain line column.

7. Checking

• Measure check lines (a line from the apex to a point on the opposite side) and tie lines; compare the measured length with the length scaled from the plot.
• Measure each line twice, forward and backward.

8. Office work

Plot the triangles to scale using arcs, plot the offsets, draw the details and ink the plan, then compute areas.

Chain surveying is best for small, open and fairly level areas.

The obstacles in chain surveying are of three types:

1. Chaining free but vision obstructed (e.g. a hill or rising ground).
2. Vision free but chaining obstructed (e.g. a pond, river).
3. Both chaining and vision obstructed (e.g. a building).

1. Chaining free, vision obstructed

Rising ground between two stations. The end stations A and B are not intervisible, but the line can be chained over the hill. The line is ranged by reciprocal ranging:

• Two persons stand at M1 and N1, points near the line such that M1 sees N1 and B, and N1 sees M1 and A.
• The person at M1 directs N1 to a point N2 on the line M1B. Then N2 directs M1 to M2 on the line N2A.
• The process is repeated until M, N, A, B are in one straight line.

       M1   _______ N1
  A ___/    hill     \___ B

2. Vision free, chaining obstructed

(a) Pond or lake on the line. Method: erect two perpendiculars AC and BD of equal length on the line at points A and B on either side; then CD is parallel to and equal to AB; measure CD. Alternatively, take a point O off the line, range A-O-C and B-O-D with OC = OA and OD = OB; the triangles OAB and OCD are congruent, so CD = AB.

(b) River. To find the width AB: at A set out a perpendicular AC, take D its mid-point, from C set CE perpendicular to AC and move along until E, D and B are in a line. Then by congruent triangles, $AB = CE$.

     B (far bank)
     :
  river
  A -----+---- D ------ C
         |             |
                      E

3. Both chaining and vision obstructed

Building on the line. Take two points A and B near the building on the line.

1. At A and B erect perpendiculars AC and BD of equal length, long enough to clear the building.
2. Join C and D and extend CD past the building. Mark E and F on line CD beyond the building.
3. At E and F erect perpendiculars EG and FH equal to AC. G and H are on the continuation of AB.
4. Distance $BG$ (hidden length) $= DE$. Measure DE and add to the chainage.

    C --------------- D ----- E ----- F
    |                 |       |       |
 ---A-------[bldg]----B       G ----- H--->

Alternatively, use angles (30° and 40° on a perpendicular) so that the line is prolonged by trigonometry.

Chain surveying is the type of surveying in which only linear measurements (with a chain or tape) are made in the field, without angular measurements. It is suitable for small, fairly flat areas with few details.

Principle

The area to be surveyed is covered by a network of well-conditioned triangles, because a triangle is the only simple figure that can be plotted from the lengths of its three sides alone. The sides (chain lines) are measured in the field. The details (buildings, roads, boundaries) are then located by measuring perpendicular or oblique offsets from the chain lines. On paper, the triangles are plotted to scale by intersecting arcs, and the details are plotted from the offsets.

        C
       /|\ 
      / | \       triangle ABC: sides measured
     /  |  \      tie line / check line shown
    A---+---B     offsets to details from AB, BC, CA

The work follows the principle of working from whole to part: the big triangles (framework) are made first and the details are surveyed from them. To avoid accumulation of errors, check lines are measured.

Ranging is the process of establishing intermediate points on a straight line between two end stations so that the chain or tape can be laid in a straight line along them. It is done with ranging rods (direct ranging) when the ends are visible, and by reciprocal ranging when they are not.

Ranging when the ends are not intervisible

When the points A and B are on opposite sides of a hill (high ground), a person at A cannot see B. In this case reciprocal ranging is used.

Procedure

1. Fix ranging rods at A and B.
2. Choose two points M1 and N1 on the high ground, close to the line AB, such that from M1 both N1 and B can be seen, and from N1 both M1 and A are visible.
3. Two assistants stand at M1 and N1. The assistant at M1 directs the one at N1 to move to N2 on the line M1B.
4. Then the assistant at N2 directs M1 to move to M2 on the line N2A.
5. The process is repeated (M2 directs N2 to N3 on M2B, and so on) until the points M and N are in the same straight line with A and B, that is, M, N, A and B are all on one line.
6. Once M and N are found, intermediate points between A and M, M and N, N and B are fixed by direct ranging.

          M1 ........ N1
        /     hill     \ 
   A  /                  \  B
   (A, M, N, B finally in one line)

The line is then chained over the high ground in the usual way.

Chain surveying is the type of surveying in which only linear measurements (with a chain or tape) are made in the field, without angular measurements. It is suitable for small, fairly flat areas with few details.

Principle

The area to be surveyed is covered by a network of well-conditioned triangles, because a triangle is the only simple figure that can be plotted from the lengths of its three sides alone. The sides (chain lines) are measured in the field. The details (buildings, roads, boundaries) are then located by measuring perpendicular or oblique offsets from the chain lines. On paper, the triangles are plotted to scale by intersecting arcs, and the details are plotted from the offsets.

        C
       /|\ 
      / | \       triangle ABC: sides measured
     /  |  \      tie line / check line shown
    A---+---B     offsets to details from AB, BC, CA

The work follows the principle of working from whole to part: the big triangles (framework) are made first and the details are surveyed from them. To avoid accumulation of errors, check lines are measured.

Operations involved in chain surveying (field procedure)

1. Reconnaissance. The area is visited and inspected, and the plan of the survey is decided.
2. Selection and marking of stations. Main stations are chosen so that lines form well-conditioned triangles; stations are marked with pegs and ranging rods.
3. Index (key) sketch. A neat freehand sketch of the area is drawn showing stations, lines and main objects.
4. Ranging and chaining. Each line is ranged between its end stations, and chained from the starting station.
5. Taking offsets. Perpendicular or oblique offsets to the details are measured with the tape and cross-staff/optical square.
6. Booking. All chainages and offsets are recorded in the field book, working from bottom to top of the page.
7. Checking. Check lines and tie lines are measured to test the accuracy of the framework.

 Recon -> Stations -> Key plan -> Chaining
      -> Offsets -> Booking -> Checks -> Office

The measurements are then plotted in the office.

Field book

The field book is an oblong book (about 20 cm $\times$ 12 cm) in which measurements are recorded in the field. It must be neat, and entries must not be erased. The records are made from the bottom of the page to the top (like the ground).

Types of field book

1. Single-line field book: A single red line (the chain line) in the middle of the page represents the chain line. The chainages are written on the line, and the offsets are written on both sides (left and right) of it.
2. Double-line field book: Two red lines run along the middle of the page, with a space between them. The chainages are written between these lines, and the offsets to the left and right are written in the columns outside, with details drawn.

 Single line type
 | left offset | chainage | right offset |
 |   12 (fence)|   60     |              |
 |             |   45     |  5 (tree)    |
 |             |   0  A   |              |

Reference sketch of stations

For each station, a small sketch (location sketch) is drawn showing its position with distances to two or more permanent objects such as corners of buildings, trees or poles.

Need:

• The station can be found again if the peg is lost.
• Stations can be re-established for checking or later work.
• It shows the name and location of the stations and lines.

Sample page of a detailing book

Chain line AB (booked upwards):

How offsets are measured and recorded

• The tape is stretched at right angles to the chain line to the object (the perpendicular by cross-staff or optical square).
• The chainage at the foot of the offset is read on the chain line.
• In the book, the chainage is written on the chain line and the offset length is written on the left or right side, with a short note (fence, tree, building) and the sketch.
• For oblique offsets, the two tie lengths are recorded along with the chainages from which they were taken.

Field work

1. Reconnaissance of the area and preparation of an index sketch.
2. Selection and marking of stations and lines so that triangles are well-conditioned.
3. Ranging and chaining of the lines.
4. Taking and measuring offsets (perpendicular and oblique) to all details.
5. Booking of all measurements in the field book.
6. Check lines and ties measured to test accuracy.

Office work

1. Check the field book, compute any corrections.
2. Choose a suitable scale and arrange the layout on the drawing sheet.
3. Plot the base line first; then plot triangles by intersecting arcs using the compass and scale.
4. Plot the offsets, and draw the details (buildings, roads, boundaries).
5. Check by scaling the check lines and compare with the field values.
6. Ink and colour the plan, add the title, north line, scale and a legend.
7. Compute the areas (triangles, offsets by trapezoidal or Simpson's rule).

Factors governing the length of offsets

1. Scale of the plan. The error in plotting is at best 0.25 mm; at a smaller scale, a longer offset can be accepted.
2. Accuracy required. High accuracy needs short offsets.
3. Method of setting the perpendicular. A tape can give a good perpendicular up to about 15 m, a cross-staff up to 15 to 20 m, an optical square up to 20 m.
4. Nature of the ground and details. Offsets to well-defined permanent details must be short; rough details allow longer offsets.
5. Obstacles and the ground. Offsets over broken or undulating ground are made shorter.
6. Importance of the object. Important features are surveyed by short offsets or ties.

Field sketch

A neat, freehand key plan (index sketch) of the area is prepared first. It shows the boundary, main details, the stations (A, B, C, D) and the survey lines with their names. The lines are numbered as they will be chained.

   Key plan (not to scale)
        C
       /|\    [House]
      / | \ 
     /  |  \     Pond  (o)
    A---D---B ---->  road
   (tie line CD, check line)

Complete process of mapping by linear measurement only

Field work

1. Reconnaissance and selection of stations A, B, C ... so that the triangles are well-conditioned (angles between $30^\circ$ and $120^\circ$).
2. Mark the stations with pegs; draw the key plan.
3. Fix the base line (longest line on level ground) and the check lines.
4. Chain the lines. Measure each side of all triangles (AB, BC, CA) and the check line CD.
5. Take offsets to the details: perpendicular offsets by tape and cross-staff or optical square; oblique offsets (ties) for hidden or far objects.
6. Book all measurements in the field book (from the bottom upwards).

Office work

1. Choose a scale (for example 1:500) and compute the length of each line on paper.
2. Plot the base line AB first. Then plot C by intersecting arcs of radii AC and BC (to scale) from A and B.
3. Plot the check line CD and compare its scaled length with the field value to detect errors.
4. Plot the offsets from the chain lines using chainages and offset lengths, and join the plotted points to get the boundaries and details.
5. Ink the plan, add north, scale, title and legend.

Thus, the whole map is made from linear measurements only, since a triangle is fixed by the lengths of its sides, and details are fixed by offsets and ties.

Chain surveying is the type of surveying in which only linear measurements (with a chain or tape) are made in the field, without angular measurements. It is suitable for small, fairly flat areas with few details.

Principle

The area to be surveyed is covered by a network of well-conditioned triangles, because a triangle is the only simple figure that can be plotted from the lengths of its three sides alone. The sides (chain lines) are measured in the field. The details (buildings, roads, boundaries) are then located by measuring perpendicular or oblique offsets from the chain lines. On paper, the triangles are plotted to scale by intersecting arcs, and the details are plotted from the offsets.

        C
       /|\ 
      / | \       triangle ABC: sides measured
     /  |  \      tie line / check line shown
    A---+---B     offsets to details from AB, BC, CA

The work follows the principle of working from whole to part: the big triangles (framework) are made first and the details are surveyed from them. To avoid accumulation of errors, check lines are measured.

Well-conditioned triangle and plotting error

A well-conditioned triangle has no angle smaller than $30^\circ$ and none greater than $120^\circ$, with an ideal shape close to equilateral ($60^\circ$ each).

The location of a plotted point depends on the intersection of two arcs. If there is an error $\delta$ in a side, the apex shifts by

$$\text{displacement} \approx \frac{\delta}{\sin\theta}$$

where $\theta$ is the angle of intersection of the arcs.

For example, with $\theta = 15^\circ$, an error of 1 mm in a side shifts the point by $1/\sin 15^\circ = 3.9$ mm, while at $60^\circ$ it shifts only 1.15 mm. Hence well-conditioned triangles keep the plotting errors small.

Chain surveying is the type of surveying in which only linear measurements (with a chain or tape) are made in the field, without angular measurements. It is suitable for small, fairly flat areas with few details.

Principle

The area to be surveyed is covered by a network of well-conditioned triangles, because a triangle is the only simple figure that can be plotted from the lengths of its three sides alone. The sides (chain lines) are measured in the field. The details (buildings, roads, boundaries) are then located by measuring perpendicular or oblique offsets from the chain lines. On paper, the triangles are plotted to scale by intersecting arcs, and the details are plotted from the offsets.

        C
       /|\ 
      / | \       triangle ABC: sides measured
     /  |  \      tie line / check line shown
    A---+---B     offsets to details from AB, BC, CA

The work follows the principle of working from whole to part: the big triangles (framework) are made first and the details are surveyed from them. To avoid accumulation of errors, check lines are measured.

Offsets

An offset is the lateral measurement from a chain line to a point of detail.

• Perpendicular offset: measured at right angles to the chain line. The right angle is set with a cross-staff, optical square, or a tape (3-4-5 method). For a short offset the foot is found by swinging the tape and taking the shortest reading.
• Oblique offset: measured at any angle, when a perpendicular cannot be taken (obstacle, long distance). It is fixed by two measurements from two points on the chain line (a tie), or by an angle and a length.

   chain line  A----+------+------B
                    |      \ 
          perpendicular     \ oblique
                  object P    object Q

Both types are booked in the field book with the chainage along the line and the offset length on the left or right of the line. Perpendicular offsets are kept short (usually within 15 m) and are preferred for accuracy.

Factors in choosing stations of a chain survey

1. Stations should be intervisible and the lines should be easy to range and chain.
2. The framework should have well-conditioned triangles (angles between $30^\circ$ and $120^\circ$).
3. The base (main) line should be the longest and on level, firm ground, running through the middle of the area, so that offsets stay short.
4. The number of stations should be minimum, for less work and fewer errors.
5. Lines should pass over level ground, avoiding steep slopes and obstacles such as ponds and buildings.
6. Provide check lines and tie lines to test and to fix details.
7. The lines should run close to the details so that offsets are short (preferably not more than 15 m).
8. Stations should be on firm, permanent places and tied to permanent objects for recovery.
9. Boundary lines and detail features may be used as survey lines.

Types of offsets

1. Perpendicular offsets: measured at right angles to the chain line, using a cross-staff, optical square or the tape (3-4-5 method).
2. Oblique offsets: measured at an angle to the chain line, usually from two points on the line (tie), when perpendicular is not possible.

By length: short offsets (up to about 15 m for ordinary work) and long offsets (for rough details).

   A------+--------+-----B
          |         \ 
      perpendicular   oblique

Factors in selecting stations and chain lines

1. Intervisibility. Stations should be visible from each other so that lines can be ranged and chained easily.
2. Well-conditioned triangles. The lines should make triangles with all angles between $30^\circ$ and $120^\circ$.
3. Base line. Choose the longest line on level ground through the middle of the area; it carries the main framework.
4. Short offsets. Lines should pass close to the details so that offsets are not longer than 15 m.
5. Minimum number of stations and lines, to reduce work and errors.
6. Ground. Avoid steep slopes, ponds, buildings and thick bushes on lines.
7. Check and tie lines must be provided.
8. Stations should be on firm, permanent ground and tied to objects.

Why a well-conditioned triangle is necessary

A triangle is plotted from the lengths of its three sides by intersecting arcs. If an angle is very small or very large (an ill-conditioned triangle), the arcs intersect at a flat angle, and a small error in a measured length moves the plotted point by a large amount:

$$\text{displacement} \approx \frac{\delta}{\sin\theta}$$

where $\delta$ is the error in the side and $\theta$ the intersection angle. For $\theta = 15^\circ$, an error of 1 mm shifts the point 3.9 mm; for $\theta = 60^\circ$ only 1.15 mm. An equilateral triangle is the best shape; angles from $30^\circ$ to $120^\circ$ are acceptable. This keeps the plotting error small and the framework accurate.

Major considerations in selecting main stations and lines

1. Stations must be intervisible and permanent, on firm ground.
2. The framework should consist of well-conditioned triangles (angles between $30^\circ$ and $120^\circ$).
3. The main (base) line should be the longest, on level ground, near the middle of the area.
4. Use the minimum number of stations and lines.
5. Lines should be close to the details so that offsets are short (preferably up to 15 m).
6. Avoid obstacles and steep slopes; do not let lines pass over ponds or buildings.
7. Provide check lines and tie lines for testing and for the details.
8. Some lines may run along fences or roads to reduce the number of offsets.

Number of offsets depending on the object

The number of offsets depends on the nature of the object and the scale of the plan.

The more irregular the shape and the larger the scale, the more offsets are needed, so that the drawn outline resembles the real shape.

Why a well-conditioned triangle is necessary

A triangle with all angles between $30^\circ$ and $120^\circ$ (ideally $60^\circ$) is called well-conditioned. The triangle is plotted from the three measured sides by intersecting arcs. In an ill-conditioned triangle (an angle very small or large) the arcs cut at a flat angle and a small error in a side moves the plotted point a lot:

$$\text{displacement} \approx \frac{\delta}{\sin\theta}$$

With $\theta = 15^\circ$, 1 mm error becomes 3.9 mm; with $\theta = 60^\circ$ it becomes only 1.15 mm. Hence the framework should be formed of well-conditioned triangles to keep plotting errors small.

Importance of the tie line

A tie line joins two fixed points (tie stations) on the main survey lines. Its importance:

1. Check on the framework. The scaled length of the tie line on the plan is compared with its measured length; a difference reveals errors in the main lines.
2. Location of details. It is used to take offsets to details that are far from the main lines, and to fix interior objects.
3. Fixing the position of the survey lines: keeps the triangles from distorting and so controls the shape.
4. Avoid long offsets. A tie line can be run through the middle of a big triangle where details are far, so that offsets remain short.
5. Joining the stations of different lines and giving a way to start a new triangle.

        C
       /|\ 
      / | \ 
     /  |  \   tie line DE
    A---D---B

Field book

The field book is an oblong book (about 20 cm $\times$ 12 cm) in which measurements are recorded in the field. It must be neat, and entries must not be erased. The records are made from the bottom of the page to the top (like the ground).

Types of field book

1. Single-line field book: A single red line (the chain line) in the middle of the page represents the chain line. The chainages are written on the line, and the offsets are written on both sides (left and right) of it.
2. Double-line field book: Two red lines run along the middle of the page, with a space between them. The chainages are written between these lines, and the offsets to the left and right are written in the columns outside, with details drawn.

 Single line type
 | left offset | chainage | right offset |
 |   12 (fence)|   60     |              |
 |             |   45     |  5 (tree)    |
 |             |   0  A   |              |

Reference sketch of stations

For each station, a small sketch (location sketch) is drawn showing its position with distances to two or more permanent objects such as corners of buildings, trees or poles.

Need:

• The station can be found again if the peg is lost.
• Stations can be re-established for checking or later work.
• It shows the name and location of the stations and lines.

Sample page of a detailing book

Chain line AB (booked upwards):

How offsets are measured and recorded

• The tape is stretched at right angles to the chain line to the object (the perpendicular by cross-staff or optical square).
• The chainage at the foot of the offset is read on the chain line.
• In the book, the chainage is written on the chain line and the offset length is written on the left or right side, with a short note (fence, tree, building) and the sketch.
• For oblique offsets, the two tie lengths are recorded along with the chainages from which they were taken.

Obstacles in chain survey and their remedies

Obstacles are of three types.

1. Chaining free, vision obstructed (rising ground). Use reciprocal ranging: assistants stand at M1 and N1, each directing the other onto the line to the far station, until M, N, A and B are in one line.

2. Vision free, chaining obstructed (pond or river).

• Pond: erect two equal perpendiculars AC and BD at the points A and B on the line on either side, measure CD, which equals AB.
• River: at A set a perpendicular AC, take D its mid-point, at C set CE perpendicular to AC and move along it until E, D and B are in line. Then $AB = CE$.

3. Both chaining and vision obstructed (building). Take A and B before the building and erect equal perpendiculars AC and BD, join CD and extend beyond the building to E and F, set equal perpendiculars EG and FH. G and H are on the original line, and $BG = DE$.

 C ------- D ---- E ---- F
 |         |      |      |
 A-----[ bldg ]---B      G---- H --->

Obstacles in chain surveying are of three types: (1) chaining free but vision obstructed, (2) vision free but chaining obstructed, and (3) both obstructed.

1. Chaining free, vision obstructed

Rising ground between two stations A and B. Use reciprocal ranging. Two assistants at M1 and N1 (each sees the other and one end station) direct each other alternately onto the lines to the far stations, until M, N, A, B are in one line. The line is then chained over the hill.

        M1 ...... N1
   A ___/  hill     \___ B

2. Vision free, chaining obstructed

(a) Pond. Erect equal perpendiculars AC and BD at the points A and B on the line on opposite sides. Measure CD; then $AB = CD$.

(b) River (width AB). At A set a perpendicular AC; take its mid-point D; at C set CE perpendicular to AC; move along CE until E, D and B are in line. Then triangles ABD and CED are congruent, so $AB = CE$.

          B   (opposite bank)
          :
   river  :
   A ---- D ---- C
                 |
                 E

(c) Distance across by angles. Set a perpendicular AC at A, measure the angle ACB; then $AB = AC \tan\angle ACB$.

3. Both chaining and vision obstructed

A building on the chain line. At A and B, close to the building, erect equal perpendiculars AC and BD, long enough to clear it. Join C and D, extend CD beyond the building to E and F. At E and F erect equal perpendiculars EG and FH. The line G-H is the prolongation of AB, and the hidden distance $BG = DE$.

 C ------- D ---- E ---- F
 |         |      |      |
 A---[ bldg ]----B       G ---- H --->

Field sketch for detail survey by linear measurements

The chain lines (framework) and objects are drawn freehand in the field book as a key plan. A detail survey by chain or tape is made by:

1. Running the main chain lines AB and BC and a check line.
2. Taking perpendicular or oblique offsets to buildings, roads, trees, fences and booking them.
3. Drawing a sketch of each object beside the chain line, with its dimensions.

        C
       /|\    [House] offset 8 m
      / | \ 
     /  |  \     (o) tree
    A---D---B ----> road edge

Later the triangles are plotted by arcs and the details by offsets.

Data

BC = 121.92 m, perpendicular to AB at B. At C, $\angle BCD = 30^\circ$ and $\angle BCE = 40^\circ$. D and E are to lie on the prolongation of AB beyond the building. Chainage of B $= 95.10$ m.

 A ------ B ---[ building ]--- D ----- E ---->
          |                  /       /
 121.92   |              /      /
          |          /     /
          C ----------------
          (30 deg at C to D, 40 deg at C to E)

Triangles CBD and CBE are right-angled at B.

Length CD

$$CD = \frac{BC}{\cos 30^\circ} = \frac{121.92}{0.8660} = 140.78\ \text{m}$$

Length CE

$$CE = \frac{BC}{\cos 40^\circ} = \frac{121.92}{0.7660} = 159.16\ \text{m}$$

Distances BD and BE

$$BD = BC\tan 30^\circ = 121.92\times 0.5774 = 70.39\ \text{m}$$ $$BE = BC\tan 40^\circ = 121.92\times 0.8391 = 102.30\ \text{m}$$

Chainage of D

$$\text{Chainage of D} = 95.10 + 70.39 = 165.49\ \text{m}$$

(Chainage of E $= 95.10 + 102.30 = 197.40$ m.)

Answer: CD = 140.78 m, CE = 159.16 m; chainage of D = 165.49 m.

Data

Points R, Q and S are collinear with RQ = 80 m, QS = 130 m, so RS = 210 m. P is on the chain line, with PR = 230 m and PS = 280 m. PQ is required.

            P
           /|\ 
      230 / | \ 280
         /  |  \ 
        R---Q---S
         80  130

Stewart's theorem

For point Q on side RS of triangle PRS:

$$PR^{2}\times QS + PS^{2}\times RQ = RS\,(PQ^{2} + RQ\times QS)$$ $$230^{2}\times 130 + 280^{2}\times 80 = 210\,(PQ^{2} + 80\times 130)$$ $$52900\times 130 + 78400\times 80 = 6{,}877{,}000 + 6{,}272{,}000 = 13{,}149{,}000$$ $$PQ^{2} + 10{,}400 = \frac{13{,}149{,}000}{210} = 62{,}614.29$$ $$PQ^{2} = 52{,}214.29,\qquad PQ = 228.50\ \text{m}$$

Check using the cosine rule

$\cos R = \dfrac{230^{2} + 210^{2} - 280^{2}}{2\times 230\times 210} = 0.19255$, so

$$PQ^{2} = 230^{2} + 80^{2} - 2\times 230\times 80\times 0.19255 = 52{,}214.29$$

Answer: PQ = 228.50 m.

A chain line crosses a river, so QR cannot be chained; it is found by trigonometry from the offset QS set out at Q (angles at S are measured with a theodolite or compass).

Given: QS = 90 m (perpendicular to PQR), PQ = 45 m, chainage of Q = 650 m, bearing of SR = 87°30′20″, bearing of SP = 177°30′20″.

    S
    |\ \
    |  \  \
 90 |   \   \
    |    \    \
 ---P-----Q-----R---  (river between Q and R)
     45 m     ?

Step 1: Angle PSR

$$\angle PSR = 177^\circ 30'20'' - 87^\circ 30'20'' = 90^\circ 00'00''$$

Step 2: Angle QSP from the right triangle SQP

$$\tan \angle QSP = \frac{PQ}{QS} = \frac{45}{90} = 0.5 \;\Rightarrow\; \angle QSP = 26^\circ 33'54''$$

Step 3: Angle QSR

Q lies between P and R, so

$$\angle QSR = \angle PSR - \angle QSP = 90^\circ - 26^\circ 33'54'' = 63^\circ 26'06''$$

Step 4: Length QR from the right triangle SQR

$$QR = QS \tan \angle QSR = 90 \times \tan 63^\circ 26'06'' = 90 \times 2.000 = 180.00\ \text{m}$$

Step 5: Chainage of R

$$\text{Chainage of R} = 650 + 180 = 830\ \text{m}$$

(Chainage of P = 650 − 45 = 605 m.)

Answer: width of river QR = 180 m; chainage of R = 830 m.