History
Surveying is the science and art of measuring distances and angles on or near the surface of the
earth. It is an orderly process of acquiring data relating to the physical characteristics of the earth
and in particular the relative position of points and the magnitude of areas. Evidence of surveying
and recorded information exists from as long ago as five thousand years in places such as China,
India, Babylon and Egypt.
Ancient Egyptian surveyors were called harpedonapata (rope-stretcher). They used ropes and
knots, tied at pre-determined intervals, to measure distances. The 3-4-5 triangle (later formalised
by Pythagoras) was discovered to give a right angle easily by using a rope knotted at distances of
3,4 and 5 units (as below) and shaped (stretched) to form a triangle with a knot at each corner
(vertex).
Surveying activity 1: An offset survey
Aim:
To practice the method of surveying using offset (perpendicular) measurements from a given base
line. To calculate perimeter and area.
What you need:
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Worksheet
Paper, ruler, pencil and eraser,
Protractor
Calculator
Instructions for students:
The diagrams on the worksheet represent notes taken by a surveyor in the field. The surveyor
began by dividing the area to be surveyed into two sections. This was the base-line and
perpendicular distances were measured and jotted down along the base-line ( the open centre-line
in the diagrams) in a notebook.
Question: What equipment do you think the surveyor has used for this survey?
Your job is to re-join the two sections and produce an accurate scale drawing of the piece of land
that was surveyed. You must select an appropriate scale for the size of your page and take care to
be accurate - use a ruler and be sure to measure or construct the right angles carefully.
When your drawing is complete find the perimeter (by measuring) and the area (by calculation) of
the land. Answer to the nearest metre.
Question: What area rules will you need? Write them down and check they are correct.
Notes for teachers:
The following worksheet has four diagrams of offset survey notes. The answers are given at the
bottom of the page.
Perimeter can be measured carefully with a ruler and should give results to within one or two
metres of the answer. The calculation of area will be more difficult requiring the shape to be divided
into component pieces. Results should generally agree with the given answers. Students could be
asked to consider the sources of error if their area does not agree.
This technique of surveying can be used to find the perimeter and area of a selected part of the
school grounds. Students should think about the equipment needed to perform the survey and
remember that good practice is to double check all measurements by an alternative means. For
example, back-bearings to check bearings, two independent measures of distance, two angles to
position a point and three to check its validity. In some schools dumpy levels are available through
the Manual Arts department and are excellent for this type of work.
Worksheet:
Assume all measurements are in metres. Choose an appropriate scale.
Answers:
1.
2.
3.
4.
P
P
P
P
=
=
=
=
199m,
178m,
202m,
269m,
A
A
A
A
=
=
=
=
266m2
1516m2
2727m2
3943m2
Surveying activity 2: Triangulation
Aim:
To practice the method of surveying using triangulation. To find perimeters and areas of the land
surveyed.
What you need:
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Worksheet
Paper, ruler, pencil and eraser
Protractor
Calculator
Instructions for students:
The diagrams on the worksheet represent notes taken by a surveyor in the field. The surveyor has
established a baseline XY of known length. W is the position to be determined and this has been
accomplished by taking bearings to W from X, and to W from Y. A rough sketch and the
measurement details are recorded in the surveyor's notebook.
Question: What equipment do you think the surveyor has used for this survey?
Your job is to produce an accurate scale drawing of the area surveyed. You must select an
appropriate scale for the size of your page. Consider the angles carefully. You may need to think
about the angles involved as each diagram has a north line. For surveying the relationship to north
is important. Take care with question 4. It will give you a four-sided figure and is an example of a
multiple triangulation (in practice surveyors may use many triangles to cover an area).
When each drawing is complete find the perimeter (by measuring) and the area (by Heron's
Formula) of the land. Question 4 will require you to think about the area calculation. A different
method may be used. Trigonometry may be required. Answer to the nearest metre.
Question: What is Heron's Formula for the area of a triangle? Write it down and remember it.
Notes for teachers:
The following worksheet has four diagrams of notes taken in survey by triangulation. The answers
are given at the bottom of the page. Encourage students to discuss the sources of error and decide
whether the errors are significant or not. What is an acceptable difference to the given answer?
This technique of surveying can be practiced in the school grounds. Students should think about the
equipment needed to perform the survey. Hand held compasses are sufficient but you may find the
Manual Arts department has a dumpy level, or two, and these are the better option. Plane table
equipment and techniques can also be used. Remember that it is good practice to double check all
measurements by an alternative means. For example, back-bearing to check bearings, two
independent measures of distance, two angles to position a point and three to check its validity.
Worksheet:
Assume all measurements are in metres. Choose an appropriate scale. Remember to work in
relationship to the indicated north line.
Answers:
1.
2.
3.
4.
P
P
P
P
=
=
=
=
135m,
124m,
207m,
235m,
A
A
A
A
=
=
=
=
820m2
718m2
2039m2
3262m2
Surveying activity 3: Radiation
Aim:
To practice the surveying technique of radiation. To find perimeter and area.
What you need:
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Worksheet
Paper, ruler, pencil and eraser
Protractor
Calculator
Instructions for students:
The diagrams on the worksheet represent notes taken by a surveyor in the field. The surveyor has
first drawn a north line to which all bearings and distances are referenced. The north line does not
form a part of the land area being surveyed. This would only be the case if there were a distance
indicated on it.
Question: What equipment do you think the surveyor has used for this survey?
Your job is to produce an accurate scale drawing of the land that has been surveyed using the
information from the notes. You must select an appropriate scale for the size of your page and take
care with the accuracy of your measurements.
When your drawing is complete find the perimeter by measuring the lengths of the sides carefully.
Also find the area of the last scale drawing using Heron's formula. Answer to the nearest metre.
Question: What is Heron's Formula for the area of a triangle? Write it down and remember it.
Notes for teachers:
The following worksheet has four diagrams of surveys by the method of radiation. The answers are
given at the bottom of the page. Students should discuss the source of errors and consider the size
of an error as significant, or otherwise.
This technique of surveying can be used to find the perimeter and perhaps the area of a selected
part of the school grounds. Students should think about the equipment needed to carry out the
practical work of the survey and remember that it is good practice to double check all
measurements by an alternative means. For example, back-bearings to check bearings, two
independent measures of distance, two angles to position a point and three to check its validity.
Investigate whether there are dumpy levels available in the school, perhaps through the Manual
Arts department. Hand compasses are sufficient but to experience using a dumpy level would be
better.
Worksheet
Assume all distance measurements are in metres an angle measurements are in degrees. Choose
an appropriate scale. Label the diagrams clearly and show all lengths.
Answers:
1.
2.
3.
4.
P
P
P
P
=
=
=
=
178m
266m
231m
253m, A = 2919m2
Surveying activity 4: In the distance
This idea was presented to me by Bob Christopherson as a way to explain in simple terms the
workings of a dumpy level. It demonstrates the principle extremely well and is a great activity to do
with students. Thankyou for this contribution.
Aim:
To make a simple distance measuring instrument and use it to approximate the distance to an
observed object. To develop an understanding of the principles by which surveying equipment such
as a dumpy level works.
What you need:
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Piece of smooth timber 480 x 50 x 10 prepared as shown below
A simple distance measuring device
Measuring tape or stadia
Chalk Instructions for students:
In this activity you may be required to make the simple equipment, or you may be supplied with
them already prepared. Examine the item and note anything important about the dimensions and
position of the triangle drawn on the wood.
Use the measuring tape to mark a distance of one metre in chalk on a vertical surface such as a
wall or post. It does not have to be from the ground but must be exactly one metre between the
two chalk marks. If using a stadia rod then just find appropriate markings on the rod for an interval
of one metre.
Now take the simple measuring device. Hold the corner of the triangle marked on the wood directly
beside your eye – if there is a hole drilled in the wood this assists levelling the triangle with the eye
because you can put your finger through the hole and hold it steady beside your eye. Move
backwards and/or forwards until the two nails line up with the one metre you have marked.
Measure the distance from the metre marking on the object to where you are standing. Record this
distance.
Change the marking on the vertical surface and repeat the process. Try two different distances (e.g.
0.65m and 1.3m). Record your distances measured on the ground from the object. Is their a
relationship between the measuring device, its triangle and nails, and the results you have
obtained?
Finally select two objects at some distance and approximate their distance from you with the
measuring device. How does this relate to the workings of a dumpy level? Explain.
Notes for teachers:
Materials for this activity are inexpensive and may be provided free of charge by your Manual Arts
department or even the students. The instruments are really simple to make and durable. Students
may be able to make them themselves or you may chose to have them already prepared, ready for
use.
Choose a fairly open area to do this work. The classroom is not really suitable as desks get in the
way and distances of more than five metres will be difficult to accommodate.
Extension:
Some golfers use a small instrument called a range finder to estimate the distance to the flag. It
works on the assumption that the flag is a fixed height of two metres. This is not actually true as
the height of the flag depends on the location of the hole, its elevation and its visibility.
The view through the range finder is shown in the diagram below.
Assuming the flag is fixed at two metres in height, students could investigate:
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How is the curve derived?
Find an equation to fit the curve using a graphics calculator, or by using algebra and a
knowledge of functions (Maths B).
Discuss accuracy and reliability of the distances approximated.
Evaluate the practical worth of the instrument to golfers
Surveying activity 5: How far?
Aim:
To use the principles of geometry and trigonometry to determine the distance between two
locations, in particular places which are inaccessible to manual measurement.
What you need:
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Measuring tape or trundle wheel
Compass or dumpy level
Notebook and pen to record data
Instructions for students:
Method (1): Imagine an object located on the far side of a river. You want to know how far it is
away. This can be done using the properties of an isosceles triangle. Fix a position at 90° to the
object on the near side of the river. Walk at 90° to this line along the bank until the bearing of the
object is 45°. Measure the distance walked. This is equal to the distance across the river to the
object. See diagram A below.
Method (2): Use trigonometry to calculate the distance across the river. Fix your position at 90° to
the object on the near side of the river. Walk at 90° to this line along the bank for a known distance
and take a bearing to the object. The tangent ratio can then be used to calculate the distance
across the river. See diagram B below.
Method (3): Take any two bearings to the object from a base-line on the near side of the river.
Measure the distance between the positions from which the bearings were taken. Produce an
accurate scale drawing from the data collected. Construct and measure the perpendicular distance
to the object on the scale drawing.
Method (4): Any other legitimate procedure. Students may find any number of ways to calculate
the distance - great! The important thing is for the solution to be supported by diagrams and
calculations which justify the method used.
This type of practical work can be done in the school grounds by selecting an object that lies
beyond some sort of barrier. For example, on the other side of a fence, road, creek or pathway. An
important rule is that no one crosses the barrier.
Students can be asked to use at least two different methods to find the distance and to justify the
work they have done with diagrams and calculations. Whatever method is used the answers
obtained should agree.