Download Surveying laws and geometry

Building Site Surveying
and Set Out
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Topics in module
1. Surveying laws and geometry
2. Survey plans, bearings and set out
3. Measuring equipment
4. Leveling devices, tools and instruments
5. Using your level and level books
6. Grid levels, areas and volumes
7. Plotting and calculating grades
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Topic 1. Surveying laws and geometry
Contents
The laws relating to surveying
 Surveyors and survey qualifications
 Cadastral surveying
 Survey marks
Geometry revisited
 The measurement of angles
 Right-angled triangles and Pythagoras’s theorem
 Triangles without a right angle
 Triangles and building sites
Learning outcomes
On completion of this section you will:
• know the various Acts of parliament which affect building setting-out
• know which areas of surveying require the services of a practicing Registered
Surveyor
• be able to identify various survey marks
• be able to calculate missing values in a right-angled triangle and use trigonometrical
ratios to calculate oblique triangles
• be able to use these triangles as an aid in setting out buildings.
Introduction
Setting out for building construction requires some knowledge of basic
surveying. Part of this section is devoted to a brief introduction to land
surveying in Australia.
Surveying has been described as the second-oldest profession. About
5000 years ago, surveyors were in great demand in Egypt to mark out the
corners of each farm allotment following the annual flooding of the land
by the River Nile. This was most important, because the Pharaohs based
taxation on the area of land owned and the estimated yield from that area
of land.
In the Book of Genesis in the Bible, Moses exhorted the Israelites, ‘Thou
shalt not remove thy neighbour’s corner mark’. Just 2000 years ago the
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Roman agrimensor (agri- for land and mensor for measurer) was in great
demand to set out villages, roads and aqueducts.
In Australia, surveying is one of the very few professions which are
controlled by government statute. There are laws for the qualifications of
surveyors, laws as to the type of surveying which only surveyors can
carry out, the type of marks which are placed and even the dimensions of
these marks.
The laws relating to surveying
As mentioned in the introduction, there are several Acts of parliament or
parliamentary statutes which affect the practice of land surveying in
Australia. Each state has its own specific laws.
Here are some of those laws which apply specifically to NSW.
The principal Act of parliament which controls the practice of land
surveying is the Surveyors Act, which sets out:
• the academic qualifications required for registration as a surveyor
• the types of surveys which only a Registered Surveyor can carry out
• the types of survey marks which are used to define a parcel of land
• the dimensions of these marks.
Some other acts of parliament which have a bearing on surveying are the
Local Government Act, the Real Property Act and the Survey
Coordination Act.
Surveyors and survey qualifications
Within any surveying organisation there are three levels:
• Registered Surveyor
• Survey technician
• Survey field hand.
Of course these three levels are not clearly defined and there are ‘grey’
areas in between each level. There are also other support staff such as
clerical workers, draftspeople and so on.
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Table 1: Three levels of surveyors’ qualifications
The Surveyors Act specifies that any firm or business setting up as land
surveyors must have Registered Surveyors as its principals.
Registered surveyors
For registration as a surveyor by the Board of Surveyors, the candidate
must have a degree in surveying from a recognised university. On
completion of the degree, the candidate must then complete two years as
an articled pupil to a Registered Surveyor. On completion of the two
years, the candidate then must pass a practical and oral examination set
by the Board of Surveyors. This final examination carries over five days.
Each year in the Government Gazette, the Board of Surveyors publishes a
list of all surveyors who are registered.
Technician surveyors
Technician surveyors are those who have completed a degree at the
university and have not for various reasons bothered with registration, or
have completed a surveying course through TAFE (Surveying Certificate
or Surveying Diploma). There are some technicians without qualification,
who through skill and expertise are competent to use surveying
equipment and carry out the necessary calculations.
Survey field hand
The survey field hand is the equivalent of the surveyor’s labourer—the
one who has to scramble over fences, is chased by savage guard dogs,
confronts irate property owners and invariably has to cut a line through
the thickest patch of blackberry bush within kilometres!
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Cadastral surveying
The term cadastral simply means boundary surveying. The Surveyors Act
states that cadastral surveys must be carried out by a Registered Surveyor
or by a survey technician under the immediate supervision of a
Registered Surveyor.
This part of the Act is very important because it means that only
Registered Surveyors or technicians under immediate supervision can
mark out the corners of a property or define the location of a boundary.
The two types of boundary survey which are of concern to us are surveys
for property identification and surveys to mark property boundaries.
Surveys for property identification
The boundaries of a property are located but not necessarily marked. The
survey identifies that the land which is the subject of the survey agrees
with the description of the land given on the title documents. Any
improvements on the property are usually related to the property
boundaries, and the survey report also shows any restrictions on the title
by way of easements or covenants, and also any encroachments on the
land or by the land on adjoining property.
Surveys to mark property boundaries
In this case the corners of the property are marked, and sometimes,
depending on the length of the boundary or nature of the boundary, line
marks are placed along the boundary. Additional line marks may also be
placed at the request of the client as an aid to building setting-out.
There are other types of surveys defined by the Act which are used for the
definition of boundaries. These are surveys to create new titles or to
amend existing property titles. An example is a subdivision survey where
new allotments are created.
It is important to note that cadastral or boundary surveys are the only
surveys which are specified by the Surveyors Act requiring supervision by
a Registered Surveyor.
Survey marks
Marking corners
The Surveyors Act specifies the type of marks to be used for marking
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corners, angles and line marks on property boundaries. For example in
NSW, for an urban holding, the dimensions of the peg marking a corner
should be 75 mm by 50 mm and 450 mm long. These pegs are driven into
the ground leaving only 75 mm of the top of the peg above ground. The
actual corner is marked on the top of the peg by a clout.
Each state has its own standard for survey marks. Become familiar with
the type of mark used m your state to ensure that you do not become
confused when setting out a building and use the wrong marks.
Where it is not possible to place a peg (for example, where the boundary
is fenced), the corner is marked by a galvanised clout or by a spring-head
roofing nail. In rock, the corner is marked by a drill-hole in the rock, with
wings cut in the rock indicating the direction of the boundary lines.
For rural allotments in NSW, the corners are marked by pegs 75 mm by
75 mm, with trenches or a line of stones two meters long on each side of
the peg to indicate the direction of the boundaries. These are called Lock
spits. The centre of the peg is taken as the actual corner.
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Figure 1 shows some of the different types of marks used for marking
property boundaries.
Other survey marks
There are many other marks specified by law which surveyors place
when carrying out surveys for new subdivisions or for redefinition of a
land title. Some of these are called permanent marks, reference marks and
state survey marks.
Reference marks can take the form of a drill and wing similar to the drill
hole and wings which indicate a property corner. Take
care if the mark appears to be ‘off-line’ to the rest of the boundary marks.
There are also two pegs which are used to mark points which are not
corners of properties. First there is the ‘dumpy’ peg. This is a short peg
the head of which is much smaller than the property corner peg.
Note that dumpy pegs are never used to mark property corners. These
pegs are not described in the Surveyors Act. The uses to which these pegs
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are used are to mark corners of buildings, centre lines of pipelines, centre
lines of roads and driveways, and anything else not related to property
boundaries. By the same token, survey pegs described by the Surveyors
Act are never used for anything else but property corners. This is to avoid
confusion about marks.
Indicator pegs are used to indicate the location of a dumpy peg, because
dumpy pegs are driven so that the top of the peg is flush with the ground.
Indicator pegs are 50 mm by 20 mm and a length sufficient so that the top
of the indicator will show above the paspalum or other grass which will
hide the location of the dumpy.
One final point before we leave survey marks: just as Moses could call
down fire and brimstone on anyone who removed his neighbour’s corner
mark, the Board of Surveyors has power under the provisions of the
Surveyors Act to impose penalties on anyone who willfully removes or
disturbs a survey mark. Not quite fire and brimstone, but a heavy fine,
and the miscreant also has to pay for the replacement of the mark in its
correct position.
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Geometry revisited
This is not going to be a detailed or heavy exposé on the proof of
theorems with which you may have battled at school. We will look at
some very basic formulae and the application of the formulae to solve
right-angled triangles. Many surveying calculations are based on simple
right-angled triangles.
Before we delve into triangles, there is one area for which clarification is
needed, the measurement of angles.
The measurement of angles
You may recall that a degree is divided into 60 minutes (‘) and each
minute is divided into 60 seconds (”). One second is one three-thousandsix-hundredth of one degree (‘1/3600°). This may seem a very minuscule
fraction, but surveyors are used to working down to fractions of this size.
In fact, the Surveyors Act states that all angles shall be shown to the
nearest five seconds.
Converting degrees, minutes and seconds into decimals
Unfortunately calculators cannot work directly in degrees, minutes and
seconds. We need to convert these fractions of angles into decimals of a
degree.
Scientific calculators have a built-in function key which allows you to
enter degrees, minutes and seconds, and this key will then convert them
into decimals.
Before you reach for your calculator to see how this function key works,
here is the simple routine to convert angles to decimal degrees:
minutes seconds
Unfortunately all calculator manufacturers seem to have their own ideas
how the angle-conversion function key should work. Refer to the
handbook which came with your calculator to find out how your
particular machine works.
Whichever way your machine requires the entry of angles, remember you
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must always convert angles from degrees, minutes and seconds to
decimal degrees before carrying out any mathematical operation of the
angles.
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Small fractions of degrees
We have already mentioned small fractions of degrees. These small
fractions also carry over when you convert to decimals. For example, one
minute = 0.0166667 in decimal degrees. One minute and 5 seconds =
0.0180556 decimal degrees.
You can see that there is a substantial difference in the third decimal
place. Your calculator will work to 7 or 8 places so always show all these
places. Do not round off to four places. You will introduce errors.
One final point when working with angles, always show high-order zeros
for minutes and seconds. That is 5 minutes is always shown as 05’ and
not 5’.3 minutes and 5 seconds is shown as 03’OS”. This is to avoid the
possibility of errors.
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Check your progress 1
Work through these problems. Convert the following angles to decimal
degrees: show your answers to seven decimal places.
1. 15°08’30” _____________________________
2. 63°27’55” _____________________________
3. 41°03’45” ___________________________
4. 88°57’05” _____________________________
5. 3°O1’50” _____________________________
6. 19°33’lO” ___________________________
7. 71°O8’25” _____________________________
8. 57°46’SO” _____________________________
9. 31°12’20” ___________________________
10. 29°55’05” _____________________________
Right-angled triangles and
Pythagoras’s theorem
Pythagoras’s theorem
Pythagoras’s theorem states that the square on the hypotenuse of a rightangled triangle is equal to the sum of the squares on the other two sides.
Useful formulae
Let’s have a look at a right-angled triangle and break this sentence down
into simple, easy-to-remember formulae.
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The hypotenuse of any triangle is the longest side.
Now let’s try an example.
Figure 4: A right-angled triangle
In this triangle we are given two sides and require to determine the length
of the third side.
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a2 =b2+c2
=42 32
= 16 + 9
=25
a=√25=5
This is the classic 3:4:5 triangle and as you can see, the calculations are
quite simple. Remember, if you are given the hypotenuse (the longest
side) and one other side, the rule is to subtract and not add.
Check your progress 2
Figure 5 shows six different right-angled triangles each with the value of
one side missing. In each case calculate the missing value.
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Figure 5: Right-angled triangles with missing values
Sine, cosine and tangent
Unfortunately the calculation of a missing side from two other sides is
only part of the story. As you may have gathered from the previous
pages, angles play an important part in the solution of triangles for
missing values.
There are three trigonometrical ratios related to right-angled triangles
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which have an important bearing on our calculations. These are the sine,
cosine and tangent ratios for angles.
Defining sine, cosine and tangent
These three ratios are:
• The sine of an angle equals the opposite side divided by (over) the
hypotenuse.
• The cosine of an angle equals the adjacent side divided by (over) the
hypotenuse.
• The tangent of an angle is equal to the opposite side divided by (over)
the adjacent side.
A simple method to remember the ratios
Note the initials/each line S = 0/H, C = A/H and T = 0/A, the three ratios.
Figure 6 illustrates the opposite and adjacent sides to angles in a rightangled triangle.
Figure 6: The sides of a right-angled triangle
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Using the ratios
How are these ratios used? In any right-angled triangle, if we are given
the value of one side and either of the two angles (note that the right
angle is always known), we can calculate all the other values in the
triangle. Let’s look at an example.
Figure 7: Right-angled triangle with missing information
In Figure 7 we are given an angle and only one side. From this
information we can calculate the other angle and the two missing sides.
The given information in this triangle is one angle and the side opposite
this angle. Given the opposite side, then the appropriate ratio to use is the
sine ratio.
sin = opposite ÷ hypotenuse
The equation needs to be rearranged to give the two known values on one
side.
Hypotenuse = opposite ÷ sin
H=28.679÷sin35°
= 28.679 ÷ 0.5735764
=50
Adjacent side = opposite ÷ tangent
A=28.679 ÷ tan35°
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= 28.679 ÷ 0.7002075
= 40.958
The adjacent side could also have been calculated from the hypotenuse
which has just been determined.
adjacent side = hypotenuse x cos
A = 50 x cos 350
=50x0.8191520
= 40.958
As you may recall from school days, the sum of all the angles in any
triangle must add up to 180°. In a right-angled triangle the right angle is
90°, so the other two angles must add up to 90°.
angle=90—35
= 55°
Now we have determined all the values off the triangle. Are you sure they
are correct? There is one final calculation which must be carried out.
Check your work!
√(28.6792 + 40.9582) = √12500
=50
This last step, checking your work, is most important. Do not neglect it.
There is nothing more disconcerting than watching the ready-mix
concrete truck drive away having delivered its load, and then discovering
that you have made an error in setting out caused by an incorrect
calculation.
Checking calculations
When checking calculations, it is not enough simply to repeat the
calculation. If you have made a mistake, then there is every likely hood
that you will make the same mistake again. If possible, always check
calculations by using a different method.
In the above calculation, notice that I have used a different method to
calculate the length of the hypotenuse as a check.
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In the initial calculation, the hypotenuse was calculated using the sine
function. After calculating the third side, the hypotenuse was recalculated using Pythagoras. This checked not only the original
calculation of the hypotenuse, but also the calculation of the third side.
Surveyors claim to never make mistakes! Of course, this is not true.
Everyone makes mistakes and surveyors try to find their mistakes before
anyone else finds them! Most survey work is either self-checking, or the
work is done using two different methods. We will come across selfchecking routines later in this module.
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Check your progress 3
Figure 8 shows four right-angled triangles, each with an angle and one
side given. Calculate the missing values for each triangle.
Triangles without a right angle
What you need to know
Right-angled triangles are a unique form of triangle. In general, most
triangles are oblique triangles. That is, the three angles are all different
and not one of them is a right angle.
The methods for calculating oblique triangles are much more complex
than for right-angled triangles and are beyond the scope of this module.
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We will come across oblique triangles a little later in another section of
this module, but even in there you will not be called upon to determine
missing values or even calculate with any accuracy.
For nearly all your surveying calculations you will be required to deal
with only right-angled triangles.
Two important rules
Just before we leave the theoretical side of triangle calculation, there are
two points you should always keep in mind. These two points apply to
any triangle whether right-angled or oblique.
• The hypotenuse is always shorter that the sum of the other two sides.
• The length of any side is always of the same magnitude of the opposite
angle.
The second rule may need a little explanation. The hypotenuse being the
largest side is always opposite the largest angle. In a right-angled triangle,
the right angle is the largest angle and the hypotenuse is always opposite
the right angle. The shortest side is always opposite the smallest angle.
Triangles and building sites
We will now apply the theory of right angle triangles from the previous
pages to building sites.
Town planners delight in creating building allotments where the shape of
the allotment is not square. The first problem is will the building as
designed fit on the block of land?
Never assume that the frontage of a block of land is the actual width of
the block. When we refer to the width of a block of land, width means the
square width, the width at right angles to the side boundary.
Defining square width
Figure 9 illustrates what is meant by the square width. Looking at the
diagram, it is obvious the length of the frontage will be greater than the
width.
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Calculating square width
Now refer to Figure 10, the same parcel of land but with some
dimensions, the angle formed by the side boundary with the frontage and
the length of the frontage. What is required is the square width of the lot.
Figure 10: Allotment frontage with dimensions
The first step in this calculation (or any calculation involving angles) is to
convert the angle to decimal degrees as shown earlier in this section.
80°47’15” = 80.7875
square width = 15.44 x sin (80.7875)
= 15.24 meters.
Taking this type of calculation one step further, you have been given a
plan of a building which the owner wants erected on his land. Figure 11
illustrates this situation. Given that the Local Government Act requires a
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minimum distance from wall to boundary on each side of the dwelling of
900 mm, will the building comply?
Figure 11: Allotment with proposed dwelling
The total width of the building is:
3500 + 5000 + 5300 = 13800 mm or 13.8 meters
71°14’ = 71.233333
square width = 16.1 x sin (71.233333)
= 15.244
available excess = 15.244 — 13.8
= 1.444 meters
the building will not comply with council requirements as 1.8 meters is
required.
Check your progress 4
Figure 12 shows a plan of land? What is the square width of the land?
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Figure 12: Land with irregular frontage
Assignment: Unit 1
Marks
1. The NSW Surveyors Act states that cadastral surveys shall be carried
out by a Registered Surveyor. Cadastral surveys include the marking of
property boundaries by pegs or other authorised means.
If you were to set up a theodolite on an existing corner peg for the
purpose of setting out a dwelling on the land, would you be
infringing the requirements of the Surveyors Act? Answer simply yes or
no, stating briefly your reasons for your answer.
2. You have been given a plan of a proposed dwelling on a parcel of land.
Given that the Local Government Act provides for a minimum wall to
boundary distance of 900 mm and the eave overhang to boundary
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distance of 675 mm, will the building comply with these requirements?
The position of the building and the pertinent lot dimensions are shown
on Figure 1 (next page).
3. Figure 2 shows a driveway onto a property. The driveway is 3 m wide
and is to be paved with concrete for the full width. From the information
given, calculate the distance apart the gateposts must be on the frontage
to clear the concrete.
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Figure 2: Plan showing location of driveway
4. Figure 3 shows the frontage and side boundaries of an irregular shaped
parcel of land. The local council has fixed a building line of 6 meters on
the land.
Determine the distance from the front corner to the position of the two
pegs marking the building line.
If the frontage of the dwelling is to be square off the northern side
boundary and the wall to boundary distances to be 1 metre, what is the
available width for the building? (The distance to be determined is shown
on the plan.)
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