Surveys and site plansmathsbooks.net/New Q Maths 12A/ch02 Surveys and site plans.pdf · SURVEYS AND SITE PLANS 37 CHAPTER 2 3 Wendel has a steady pace and knows that he takes 80 paces

Contents2.1 Review of surveying techniques2.2 Areas of irregular regions2.3 Surveying around obstacles2.4 Triangulation2.5 Open and closed traverses2.6 Plane table surveying

Chapter review

2

Surveys andsite plans

Syllabus subject matterSyllabus

guide chapter 2

Maps and compasses—land measurement

■

Drawing and interpreting site plans

■

Position fixing using directions, and vertical and horizontal measurements in relation to a datum

■

Calculation of grades (gradient of the land)

34

NEW QMATHS 12A

2.1 Review of surveying techniques

You saw in Year 11 that surveying techniques are used to prepare maps for use in construction and landscaping, land subdivision, mining, navigation and mineral exploration.

Surveyors use these techniques to:• find the relative positions of points on the Earth’s surface so that they can be shown on

maps and plans;• locate the actual positions of points on the ground—for example, find the boundary of a

block of land.

In Year 11 you looked at a number of techniques associated with surveying:• using

pacing

to measure distances• conducting a

chain survey

•

ranging

a survey line—that is, setting out a straight line using ranging poles• locating the positions of features using

perpendicular offsets

and

ties

• using a

compass

to find

bearings

and

back bearings

to plot a course or locate a feature• conducting an

open traverse

• using

levels

to find the rise or fall in relation to a

datum point

.

In addition, you

booked

the details of a survey using the same methods as surveyors. You should refer back to this material if you are uncertain about the techniques you have learnt.

People use the outputs of surveying for a variety of purposes.

Here is a list of people who may have to use surveys or the results of surveys:

• builder • architect

• geologist • land developer

• prospector • tourist

• land buyer • judge

• farmer • historian.

For each person listed, write down at least one purpose for which surveying could be important.

Visit the website of the Institution of Surveyors (www.isaust.org.au or www.isaqld.org.au for the Queensland division) to find out more about the different things that surveyors do and the various ways in which surveying is used.

Investigation Uses of surveys

Extra materialSurveying techniques

Land is a valuable commodity, and even the earliest times show that people have always needed to measure and map the land around them. Surveyors are the experts in land measurement, and their services are used for a variety of purposes beyond the preparation of site plans and maps. They may also be involved in helping to predict earthquakes, assisting police at crime scenes, monitoring and recording environmental change, and even constructing images of the ocean floor.

SURVEYS AND SITE PLANS

35

CHAPTER 2

You have used an

adjustable dial

(or

orienteering

)

compass

to find directions or

bearings

. When a bearing is less than 180

°

, add 180

°

to find the back bearing. If the bearing is greater than 180

°

, subtract 180

°

.

Builders often use a

dumpy level

to determine the slope of a building block. If a flat pad is required, earth is usually cut from the high side of the block and used to fill the low side.

Felicity takes 32 paces to cover a distance of 28 m. She finds that a rectangular sports field measures 255 paces by 190 paces. Calculate the dimensions of the sports field and thus the area of the field.

Solution

Calculate pace length. Pace length =

Substitute known values. =

Evaluate. = 0.875 m/paceCalculate dimensions of field. Field length = 255 paces

= 255 paces × 0.875 m/pace≈ 223.1 m

Field width = 190 paces × 0.875 m/pace≈ 166.3 m

Calculate area of field. Area of field = length × widthSubstitute known values. ≈ 223.1 m × 166.3 mEvaluate and round off. ≈ 37 100 m2

number of metresnumber of paces

---------------------------------------------

28 m32 paces---------------------

Example 1

a The bearing of GH is 143°; what is the back bearing?b What is the back bearing of AB in the diagram

shown at right?

Solutiona The bearing is less than 180°, so add 180°. Back bearing of GH = 143° + 180°

= 323°b The bearing is greater than 180°, so

subtract 180°.Back bearing of AB = 244° − 180°

= 64°

N

244°A

B

Example 2

36

NEW QMATHS 12A

A builder’s level is set up between two levelling staffs as shown.

Calculate the fall between the staffs and hence calculate the angle at which the ground slopes to the horizontal.

Solution

Draw a triangle to find the angle.

Mark the information onto the triangle.

Find the fall by subtraction. Fall = 2.437 − 0.243 m= 2.194 m

Use the tan ratio. tan θ =

Substitute. =

Keep the exact number on your calculator. = 0.1336 …Use tan−1 on your calculator. θ = 7.6106 …°

Round off and write the answer. The ground slopes downwards at an angle of approximately 7.6°.

ForesightBacksight

Line of collimation2.4370.243

Level

Staff

Staff

16.420 m

2.194 m

16.420 mθ

oppositeadjacent----------------------

2.19416.420----------------

Example 3

1 Calculate the pace length for each of the following people.a Will takes 40 paces to cover 42 m.b Terri takes 54 paces to cover 45 m.c Haung covers 38 m using 41 paces.d Ali covers 68 m using 77 paces.

2 Anna takes 75 paces to cover 60 m. Calculate the following distances she paced out.a 34 paces along her front fenceb 21 paces along the side of her housec 88 paces from the school canteen to her Maths roomd 688 paces from the video shop to her house

Exercise 2.1 Review of surveying techniques

SURVEYS AND SITE PLANS 37 CHAPTER 2

3 Wendel has a steady pace and knows that he takes 80 paces to cover 72 m. Calculate the area of each of the following figures paced out by Wendel.(Hint: The CD-ROM contains the formulas for areas of common shapes.)

4 a Use a protractor to find the bearings of AB, BC, CD and DE in the diagram shown here.b Calculate the back bearing of each leg.

5 What is the back bearing for each of the following?a 106° b 221° c 009° d 312° e 094°

6 A builder uses a dumpy level to find the levels at opposite corners of a proposed building block 35 m apart (measured horizontally). Measurements of 3.876 m and 1.005 m are recorded.a What is the fall across the block?b At what angle does the land slope?c Express the fall as a ratio in the form 1 : x.

a b

c d

48 paces

45 paces

66 paces

32 paces

45 paces

24 paces

15 paces

87 paces

N

A

B

C

D

E

Extra material

Area formulas

38 NEW QMATHS 12A

7 What fall is required for guttering that runs 20.4 m down one side of a house if a fall of 1 : 400 is needed to ensure that water drains sufficiently?

8 The following sketch (not drawn to scale) shows the measurements in metres taken at a construction site.

a How far above point A is point B?b What is the average angle at which the ground slopes upwards from A to B?c How far above point A is point C?d What is the average angle that the slope makes with the horizontal from A to C?e What is the average angle at which the ground slopes downwards from C to B?

Modelling and problem solving9 Trish wants to buy the block of land on the

corner of Moreland Avenue and Kenland Drive. It is advertised for $125 000. She visits the block and makes the sketch at the right, knowing that her pace length is 0.8 m.When she visits the block, Trish finds out that there is a planned easement, which means that some of the land will be unusable. When she points this out to the real estate agent, she is offered the block at $114 000 to compensate for the easement.Work out if this is a fair offer, giving reasons for your conclusion.

10 A couple are thinking about buying Lot 105 Harbour Lane to build their house. They ask their builder to inspect the block to see if it will be suitable for the style of house they wish to build. To suit their purposes, the block must have a slope diagonally across of less than 5°. The builder visits the block with a dumpy level and makes the sketch at the right.The builder knows that his pace length is 0.9 m. The dumpy level readings taken from point C are 1.073 m (at A) and 4.633 m (at B).Will the slope across the block (from A to B) be suitable for the house they wish to build?

2.831

A

B

0.927

4.612 1.203

C

24.030 20.962Not drawn

Staff

Staff

Staff

to scale

Kenland D

rive

Moreland Avenue

52 paces

5 paces

40 paces

42 pacesEasement

Elev = 4.633 m

B

AElev = 1.073 m Harbour Lane

C

Lot 105

45 paces12---


11 Go onto your school oval or any other large area of relatively flat land. Mark out five positions (separated by at least 20 paces) forming a pentagon.a From each marker, find the bearings of the other markers. Record the information

on a sketch showing the positions of the markers.b Calculate the back bearing of each bearing recorded.c On your sketch, divide the area into three triangles. Use pacing to determine the side

lengths of the triangles and hence calculate the area of the pentagon.

12 This site plan shows a house on a suburban block of ground. The plumber has paced out the distance from the corner of the house to the corner of the property where the storm water connection is located. The plumber knows that his pace length is 0.85 m.The storm water drain must be at least 1 m below the surface level and a fall of at least 1 : 50 for correct drainage.

If the drain is 1 m below the surface atthe corner of the house, at what minimum depth will the drain meet the storm water connection?

Roa

d Flatterrace

Fence

Fence

Dwelling

Storm waterconnection

37 paces

Storm waterdrain

Did you know?When taking levels over long distances, the curvature of the Earth’s surface can significantly distort measurements that are taken. As shown in the following diagram, the Earth’s curved surface departs from the horizontal line of sight by the distance ED.

In addition, the density of the air causes the line of sight to bend back or refract towards the Earth. This reduces the effects of curvature of the Earth’s surface by about 14%. The approximate effects of curvature of the Earth’s surface and refraction due to air density are much more important as the sighting distance increases, as can be seen in the table below.

Effects of curvature and refraction

Distance (m) 100 200 500 1000 3000

Error (cm) 0.08 0.38 1.62 6.88 59.38

Line of sight

Horizontal lineAssume 90°

Level surface

Level surfaceCurved surface

of the Earth

A B

D

EF

G

C

40 NEW QMATHS 12A

2.2 Areas of irregular regionsIt is important to be able to measure length so that the area of a piece of land can be calculated. The unit used for area measurement depends on the size of the land measured. Suburban building blocks, for instance, are measured in square metres. Metric units for land area are:

Other measures used for length and area of land are given on the CD-ROM.

There are many ways of calculating area. You have previously seen that when a large straight-edged area must be calculated, it may be divided into a number of smaller regions. The areas of the smaller regions are then calculated and added to give the area of the entire region.

Land area metric units1 square metre (m2) = 1 m × 1 m = 1 m2

1 hectare (ha) = 100 m × 100 m = 10 000 m2

1 square kilometre (km2) = 1000 m × 1000 m = 1 000 000 m2

!Extra

materialLand

measurement units

Calculate the area of this reserve using the field measurements shown on the diagram.

SolutionThe reserve can be divided into a triangle (�AEB) and a trapezoid (ACDB).

To find the area of �AEB we need to use Heron’s Formula.

Area =

where s = (a + b + c)

Calculate s for �AEB. s for �AEB = (45 + 57 + 92)

= 97 m

Substitute into the formula. Area of �AEB =

= Evaluate. = 1004.3 … m2

Write the rule for area of a trapezoid. Area = (a + b) × h

Substitute into the formula. Area of ACDB = (92 + 64) × 28

= 2184 m2

D B28 m

64 m

E

A

C

45 m

57 m

92 m

s s a–( ) s b–( ) s c–( )12---

12---

97 97 45–( ) 97 57–( ) 97 92–( )

97 52×××× 40×××× 5××××

12---

12---

Example 4


Regions of land whose areas need to be measured are often irregular or have curved boundaries, such as the region bounded by the river, roads and wall shown here.

In the field, such boundaries are sketched by using a survey line (AB) and a number of equally spaced offsets from the survey line to the irregular boundary. Nine offsets have been used in this example. The irregular region between AB and the river can be found by examining the ten subregions formed by the offsets, the left and right boundaries, the river and the survey line AB.

The area of this irregular region could be calculated by treating each of the subregions individually. For example, WXYZ could be treated like a trapezoid, with the offsets WZ and XY the parallel sides and WX the perpendicular distance between them. In this way,

Area of WXYZ ≈ (WZ + XY) × WX

This could be done for each subregion and then the sum of all ten areas found—but this would be time-consuming.

Find area of reserve. Area of reserve = � AEB + ACDB= 1004.3 … + 2184

Evaluate and round off. ≈ 3188 m2

Wall

Offsets

Road

River

AW

Z Y

Bridge

XB

Roa

d

12---

AW

Z Y

XB

ddddddd d d d

h1 h2 h3 h4 h5 h6 h7 h8 h9 h10

42 NEW QMATHS 12A

You can see that (WZ + XY) is the average of the offsets WZ and XY and can be representedas shown on the previous page by a dashed line (h3). The average of the offsets is known as the mid-ordinate, and h1, h2, h3, … , h10 are all mid-ordinates.So the area of WXYZ ≈ h3 × d, and the total area of the irregular region is found in the following way:

Area ≈ h1d + h2d + h3d + h4d + h5d + h6d + h7d + h8d + h9d + h10d = d(h1 + h2 + h3 + h4 + h5 + h6 + h7 + h8 + h9 + h10)

=

In fact, 10d is the length of the survey line AB. This means that an irregular region similar to the one shown here can be calculated using the mid-ordinate rule.

In the field, the surveyor establishes the survey line (AB) and measures it. The length is then divided by 10 and the first offset is taken at half this distance from A. The remaining offsets are taken at distances equal to one-tenth of AB. In this way, the offsets that are taken are actually the mid-ordinates and can be used directly to calculate the area of the region.

12---

Divide by 10 to find the average of the heights.

10d h1 h2 h3 h4 h5 h6 h7 h8 h9 h10+ + + + + + + + +( )

10--------------------------------------------------------------------------------------------------------------------------------------

Mid-ordinate ruleArea of region ≈ length of survey line × average of mid-ordinate lengths

!

Here are the field notes for the survey of a region between a fence and a line of vegetation. Use the information to find the area of the region. (All measurements are in metres.)

SolutionDetermine the mid-ordinate distance from the diagram.

Distance between mid-ordinates= 8 m

Calculate the average length of the mid-ordinates.

Average length of mid-ordinates

= Evaluate. = 22.6 mWrite the mid-ordinate rule. Area of region

≈ length of survey line × av. of mid-ordinate lengthsSubstitute values and evaluate. = 80 × 22.6 = 1808 m2 State the result. The area of the region is about 1808 m2.

15 12 22 28 27 29 24 26 29 14+ + + + + + + + +10

--------------------------------------------------------------------------------------------------------------------------

Example 5


In Year 11 you saw how the trapezoidal rule can be used to find irregular areas.

Trapezoidal ruleThe area of an irregular shape can be approximated by dividing the shape into strips of equal width, as shown.

The following rule comes from the rule for the area of a trapezium, A = (a + b) × h.

Area ≈ (s1 + 2s2 + 2s3 + … + 2sn − 1 + sn)

≈ × [(total of end lengths) + 2 × (total of middle lengths)]

≈ width × [(total of end lengths) + 2 × (total of middle lengths)] ÷ 2

s1 s2 s3 s4 s5

w

sn − 2 sn − 1 sn

12---

w2----

width2

-------------

!

Find the area of this region. (All measurements are in metres.)

SolutionThe last length in this case, after the 6 m, is 0 m. The width of each strip is 5 m.

Calculate total of end lengths. Total of end lengths = 15 + 0= 15 m

Calculate total of middle lengths. Total of middle lengths = 16 + 13 + 9 + 8 + 11 + 11 + 6= 74 m

Use the trapezoidal rule. Area ≈ (15 + 2 × 74)

=

Evaluate and round off. ≈ 410 m2 State the result. The area of the region is approximately 410 m2.

0 5 10 15 20 25 30 35 40

15 16 13 9 8 11 11 6

52---

5 163××××2

------------------

Example 6

44 NEW QMATHS 12A

Exercise 2.2 Areas of irregular regions1 In each of the following regions, all offsets are perpendicular to the survey line and

all measurements are in metres. Calculate the area of each region.

Modelling and problem solving2 A grassed area to the side of a road has been surveyed and the following perpendicular

offsets have been taken. From this field diagram, calculate the area of the grassed region using the trapezoidal rule. (All measurements are in metres.)

3 A survey is conducted of a paddock (ABCDEF) bounded by a stream on one side. Use the field measurements shown in the sketch to calculate the area of the paddock. (All measurements are in metres.)

a b

c d

24 35

41

32

68

22

24.2

18.1

26.3

61.3

38.9

19.1

78.142.5

98.2

51.3 57.4

47.4

23.733.1

26.2

38.536.8

18.3

17.1

Additional exercise

2.2


4 Here are the field notes from the surveys of two separate regions with irregular boundaries. Calculate the area of each region using the mid-ordinate rule. (All measurements are in metres.)

5 A grassed area is bounded by two intersecting roads and a wooded area. Part of the grassed area is divided into strips of 50 m width measured from a survey line, as indicated in the sketch below. Find the area of the grassed area. (All measurements are in metres.)

a b

46 NEW QMATHS 12A

2.3 Surveying around obstaclesSometimes unavoidable obstacles or obstructions are met when measuring or ranging a survey line. Some obstacles can be measured around but others cannot. We will begin by examining those that can be measured around, such as the pond that is obstructing the survey line AB shown below.

One way to measure around the pond is to make an offset, XY, near the edge of the pond. YZ is then measured back to the survey line, meeting it at Z. This forms the right-angled triangle XYZ.

We can now use Pythagoras’s Theorem to calculate the distance ZX.

6 The field notes shown here relate to a reserve bordering an industrial area. Find the area of the reserve. (All measurements are in metres and strips are 10 m wide.)

B A

Pond

B A

Pond

X

Y

Z

Find the distance ZX above if XY and YZ are measured to be 18.4 m and 65.2 m respectively.

SolutionWrite Pythagoras’s Theorem for �XYZ. YZ2 = ZX2 + XY2 Substitute known values. 65.22 = ZX2 + 18.42 Rearrange. ZX2 = 65.22 − 18.42 Evaluate. ZX2 = 3912.48Take the square root and round off. ZX ≈ 62.5 m

Example 7


Another way around the obstacle caused by the pond is to take perpendicular offsets. After the offsets PQ and SR are ranged (that is, set out as straight lines using ranging poles), RQ can be measured. RQ is the same as the required distance, SP.

Another alternative is to use similar triangles to calculate the required distance. This is done in the following way. Point C is set at a convenient place on the survey line. Next, a point E is set at some distance so that CE is clear of the obstruction. Point G is then set on the survey line at a position so that GE also is clear of the obstruction. Distances GE and CE are carefully measured and bisected at points F and D respectively.

This means that �FED and �GEC are similar. In similar triangles, the ratios of corresponding sides are equal. In this case,

= = =

So GC = 2FD, which means that the required distance is twice the distance FD, which can easily be measured.

Surveyors also use this procedure as a method of setting out parallel lines.

B A

Pond

P

Q

S

R

B A

Pond

C

D

G

F

E

FEGE-------- DE

CE-------- FD

GC-------- 1

2---

1 Form groups of three or four and use your knowledge of geometry to explain why �FED and �GEC are similar in the above case.

2 Now use geometry to explain why FD and GC are parallel.

3 Go outside to a reasonably flat area and take four or five pieces of straight timber atleast 1200 mm long (ranging poles), a spirit level, a 20 or 30 m tape, some markers anda hammer.You should refer to the investigation ‘Ranging’ on the CD-ROM if you need to be reminded about how to range a line.

4 Once outside with the equipment:• select two points that have an obstacle between them• use each of the three methods described to measure around the obstacle• compare the results obtained using the different methods.

Investigation Obstacles

Extra materialSurveying techniques

48 NEW QMATHS 12A

Sometimes the nature and size of the obstruction are such that it is not possible to measure around it. Examples of this type of obstacle include rivers and wide cuttings for roads or railway tracks, as shown below.

To measure around this obstacle, a perpendicular offset is made from a convenient point, P, out to Q. Then another perpendicular offset is made, from Q in the direction of S. Now PQ is measured and bisected at R. A convenient point, T, is chosen near the obstacle and a line ranged using T and R to locate S.

From the diagram it is clear that �TPR and �SQR are congruent. This means that the required distance (TP) is the same as QS, which can easily be measured.

B A

Road

Cutting

Cutting

B A

SQ

R

PT

Modelling and problem solving1 A wooded area falls in the path of a survey. Find the distance OM, which is needed for

completion of the survey.

2 A river obstructs the survey line XY. Find the required survey length AB using the information in the diagram.

Q PO M

N

104.5 m26.3 m

Y X

C

E

DBA

32 m46 m

65 m

River

Exercise 2.3 Surveying around obstacles


3 A railway cutting runs across a survey line, AB. Calculate the missing distance UW using the survey diagram below.

4 A large pond obstructs the survey line AB. Work out the missing length QP.

5 A gravel pit lies across the survey line AB. Calculate the missing length ML.

A

XY

Z

WU

51.2

29.4

B A

XY

Z

WU

51.2 m

29.4 m

Pond

A B

P RQ

T

S

5.0 m

3.0 m

14.3 m

A B

J

H

K

L M

29.6 m

Gravel pit

50 NEW QMATHS 12A

2.4 TriangulationIn practice, it is possible to divide most large areas of land into triangles. Triangulation is the method of surveying using triangles. Consider the following property that has to be surveyed. (All measurements are in metres.)

You can see that there are four landmarks that stand out: footbridge, street sign, pine tree and gate. So points A, B, C and E respectively should be used as survey stations. A survey peg needs to be set out at D in order to cover the entire area.

The line BE, from the footbridge to the gate, will be used as the base survey line, as point E can be used to divide the survey area into well-conditioned triangles. The well-conditioned triangles �ABE, �BEC and �EDC all have angles greater than 30°. Measurements to other survey stations (BA, EA, BC, EC, BD and ED) are chained from this line and booked. The distance DC is also chained and recorded as a check measurement. The perimeter of the survey area (BC, CD, DE, EA and AB) is measured with all chainages and offsets recorded.

Details of the survey including chainages and offsets are recorded as follows (in metres.)

Street sign

GateEA

BC

DSurvey peg

Road

Tra

ck

FootbridgeStream

Hed

ge

43 54

4864

93

11582 73


The steps in drawing the plan of this survey area are as follows:

1 Draw the base line BE to scale.

2 Use a pair of compasses set to 54 (from E) and then 115 (from B) to locate D.

3 Locate A and C in a similar way.

4 Use the offsets from BC, CD, DE, EA and AB to locate the various survey features.

B

DE

Modelling and problem solving1 Use the field notes from the example above to draw a plan of the property using the

scale 1 : 500.

2 A rectangular park contains a boating lake and sports complex, as shown below.

Road

Sportscomplex

Boatinglake

950 m

970

m

A B

C

1000 m

Exercise 2.4 Triangulation

52 NEW QMATHS 12A

A survey is conducted and the field notes of the survey are booked as follows (in metres).

Use an appropriate scale to make a scale drawing of the park, showing the features.

3 The field notes below and on the next page were completed in the survey of a factory site and storage area.

a Use the field notes to draw a rough sketch of the entire site that was surveyed.b Use the field notes and rough sketch you have just completed to make a scale drawing

of the entire site. A scale of 1 : 1000 should be used.


c Calculate the area covered by the factory.d Calculate the area of the path that services the factory.e Calculate the approximate area of the entire site that was surveyed.

54 NEW QMATHS 12A

4 The following field notes were completed in the survey of a small dairy farm.

Use a suitable scale to construct a scale drawing of the dairy showing all features.


2.5 Open and closed traversesOpen traversesIn Year 11 you saw how an open traverse is conducted using a consecutive series of lines or legs beginning from a known point and finishing at a point whose approximate location is known. The legs are chained and their directions fixed using a compass or theodolite.

The following diagram shows the proposed path for a new road connection.

The stations B, C and D are chosen carefully to ensure the best possible route for the connection road. The traverse is completed by taking forward and back bearings of AB, BC and CD and chaining the distances as indicated below.

The details of the traverse and the way it would be booked are as follows.

Leg Distance (m) Bearing

AB 642 062°

BC 1089 091°

CD 724 112°

Beginning

River

Quarryof proposedconnection

Existingroad

Connectionpoint

Exis

ting

high

way

A

B

D

C

Finish

Hig

hway

River

Quarry

Roa

d

Start

062°

091°112°

642 m

1089 m

724 m

N

56 NEW QMATHS 12A

The column on the left is used to record the bearing and back bearing of each leg, and the column on the right is used for chaining data (leg distances).

If forward and back bearings differ by exactly 180°, they are considered to be correct. However, if the difference is not 180°, it can be assumed that some local attraction that alters the bearing must exist at one or both of the stations. This needs to be corrected for the survey to be accurate.

Back

Length of AB

Bearing of AB

bearing of AB

A quick way to check bearings and back bearings is to compare the sums of the hundreds and tens digits for each. Look at the following:

125° and 305°: 1 + 2 = 3 + 0046° and 226°: 0 + 4 = 2 + 2271° and 091°: 2 + 7 = 0 + 9

However, as you can see by the next example, this equality doesn’t always work.294° and 114°: 2 + 9 ≠ 1 + 1

But by using the sums and adding again we have:11° and 02°: 1 + 1 = 0 + 2

1 Find other similar examples (between 0° and 360°) where this equality doesn’t work with the hundreds and tens digits.

2 Write a short report that summarises your findings about this procedure for checking bearings and back bearings.

Investigation Bearings and back bearings

Here are the details of an open traverse survey. Chained lengths are in metres.

Correct the bearings for local attraction.

Leg Forward bearing Back bearing

PQ 141° 317°

QR 056° 236°

RS 123° 304°

ST 014° 191°

Example 8


Solution

Draw a sketch of the traverse.We can see the forward and back bearings for each leg displayed in the table.

This is 4° less than the expected value (180°).So, there must be a local attraction acting at either P or Q. We don’t know at which station the local attraction is acting, so we need to investigate further.

Because the difference between the bearing and back bearing is 180°, it can be assumed that there is no local attraction at either Q or R. This means that the local attraction on leg PQ must be at P.

We have already seen that there is no local attraction at R, so the forward bearing of RS can be assumed to be correct. This means that the back bearing of RS needs to be adjusted.

The full table of bearings and back bearings, including all adjustments, is as follows.

Start with the first leg, PQ. Forward bearing = 141°Back bearing = 317°

Find the difference between bearings. Back bearing of PQ − bearing PQ = 317° − 141°= 176°

Examine the second leg, QR. Back bearing of QR − bearing QR = 236° − 056°= 180°

The bearing PQ must be adjusted because the local attraction acts when the compass is placed at P.

Adjusted bearing of PQ = 141° − 4°= 137°

Repeat the procedure for leg RS. Back bearing of RS − bearing RS = 304° − 123°= 181°

Adjust the back bearing. Adjusted back bearing of RS = 304° − 1°= 303°

Original Corrections Corrected

Leg Forward bearing

Back bearing Difference Forward Back Forward

bearingBack

bearing

PQ 141° 317° 176° −4° 0° 137° 317°

QR 056° 236° 180° 0° 0° 056° 236°

RS 123° 304° 181° 0° −1° 123° 303°

ST 014° 191° 177° −1° +2° 013° 193°

N

P

Q

R

S

T

304°

191°

14°

123°

236°

317°

56°

141°

441 m

335 m 312 m

438

m

58 NEW QMATHS 12A

Closed traversesWhen the legs of a traverse form a closed figure, the survey is called a closed traverse. This type of traverse is used in situations when it is not possible to chain check lines because of obstructions such as large bodies of water, buildings, woods and so on. When plotting a closed traverse, the surveyor sometimes finds that the finish and start points do not exactly coincide.

When the traverse does not close, adjustments need to be made.

The following scale drawing shows a closed traverse survey that does not close. Redraw it so that it does close.

SolutionStart by opening out the legs of the traverse to form a straight line, marking on the distances AB, BC, CD and DA1. At A1, draw a perpendicular line (A1A2) equal in length to d, the closure error.

Now join A to A2 and draw in perpendiculars from B, C and D to meet AA2 at B1, C1 and D1 respectively.

Return to the scale drawing and join AA1.

Next draw lengths equal to BB1, CC1 and DD1 from the points B, C and D respectively, making sure that they are all parallel to AA1.

The survey is now shown by the figure AB1C1D1.

A

B

CD

d

A1

A B C D

B1C1

D1A2

A1

d

A

B

C

D

d

A1

B1

C1

D1

Example 9


This way of adjusting a closed survey is called Bowditch’s method and it is a way of averaging out the errors over the entire survey.

When a closed traverse contains errors due to local attractions, these are corrected in a similar manner to open traverses.

A closed traverse PQRSP is shown opposite with bearings and back bearings indicated.

Check the forward and back bearings and make any necessary adjustments to the bearings shown.

SolutionFirst construct a table that shows the forward and back bearings of each leg.

This means that bearings taken from P, Q and S are assumed to be accurate. The remaining bearings are adjusted in the following way.

Begin with PQ. Back bearing − forward bearing = 243° − 063°= 180°

Check SP. Forward bearing − back bearing = 340° − 160°= 180°

Original Corrections Corrected

Leg Forward bearing

Back bearing Difference Forward Back Forward

bearingBack

bearing

PQ 063° 243° 180° 0° 0° 063° 243°

QR 160° 342° 182° 0° −2° 160° 340°

RS 283° 105° 178° +2° 0° 285° 105°

SP 340° 160° 180° 0° 0° 340° 160°

N

243°

160°

063°

160°

283°

342°340°

105°

P

Q

R

S


PQ 063° 243°

QR 160° 342°

RS 283° 105°

SP 340° 160°

Example 10

60 NEW QMATHS 12A

Modelling and problem solving1 These field notes show the details of part of a

property boundary survey. Chained lengths are in metres.a Use the notes and a scale of 1 : 1000 to

make a sketch of this portion of the property boundary.

b Calculate the bearing and distance of TP.

2 A traverse was made around a factory. The results were recorded as shown here. Chained lengths are in metres. Make a scale drawing of the traverse and work out the distance of leg DA.

3 The following bearings were noted in the course of an open traverse.

a Look at the table and find any legs where there is no evidence of local attraction.b Make corrections to all forward and back bearings as necessary. Make a record of these

in a new table.c Use the information contained in the table that you have just completed to make a scale

drawing of the traverse.d Find the distance and bearing of the finish point from the starting point.

Leg Forward bearing Back bearing Distance (km)

AB 052° 232° 30

BC 127° 305° 28

CD 084° 267° 42

DE 221° 043° 46

EF 119° 297° 53

FG 042° 220° 62

Exercise 2.5 Open and closed traversesAdditional exercise

2.5


4 A closed traverse is conducted to determine the position of a sewerage line in a new housing estate. The details of the traverse are shown at right.

Plot the survey using a scale of 1 : 1000. Then adjust the survey using Bowditch’s method.

5 A closed traverse is completed and the bearings are recorded in the table below. Calculate the corrected bearing for each leg and state where any local attractions appear to occur.

6 This data relates to a closed traverse. The bearings have been corrected for local attractions.

a Make a scale drawing of the survey area. You need to use back bearings for TP and ST.b Use the scale drawing to find the unknown distances in the survey.

7 The diagram below shows the detail of a closed traverse ABCDEA of an area of land. Calculate the bearing and back bearing for each leg of the traverse.

8 The results of a closed traverse (corrected for local attractions) are reported as follows.

Plot the survey using a scale of 1 : 2000. Then adjust the survey using Bowditch’s rule.


AB 144° 324°

BC 070° 253°

CD 017° 194°

DE 319° 138°

EF 228° 050°

FA 278° 098°

Leg Distance (m) Forward bearing

PQ 172 014°

QR — 320°

RS — 347°

ST 150 005°

TP 979 168°

Leg AB BC CD DE EA

Bearing 029° 101° 146° 243° 275°

Distance (m) 167 124 125 103 181

Leg AB BC CD DA

Bearing 007° 081° 163° 262°

Distance (m) 101 77 107 114

N

144°

125°

146° 138°

51°

A

B

C

DE

62 NEW QMATHS 12A

2.6 Plane table surveyingA quick and relatively easy method of conducting a survey is by using a plane table. Plane table surveying is a method of surveying in which the survey is drawn directly onto a piece of paper held in place on a portable drawing board. The plane table consists of a flat board fitted to a tripod. An example of a commercially produced plane table is shown here, but you can use any flat surface such as a small table or stool as long as you can make it level and you can fix some paper on top of it.

You need to have a sighting device known as an alidade. An alidade is used to locate points in the field. You can make one from a ruler with a nail or drawing pin fixed at each end or you can make a more elaborate device. Commercially produced alidades are also available.

In plane table surveying, the drawing is plotted line-by-line and point-by-point as the field work is completed. This means that almost all the drawing is completed while in the field and there is little need for survey notes.


Three main methods are used with the plane table: they are the radiation, intersection and traverse methods as explained below.

Radiation methodThe radiation method is used:

• where a single sighting station is sufficient to view all the required points; and

• it is convenient to chain to all required points from a single station.

1 Select a convenient area in the playground or school oval to be surveyed. Make sure that there are identifiable features on the boundaries of the area.

2 Look around the area to be surveyed and select a point (X) that is fairly central and from which all required points can be viewed and measured.

3 Mark this point on the ground with chalk.

4 Fix a piece of paper to the top of the plane table, and on it mark the position of the point you have just selected. Call this point x. Insert a pin at x. The sighting instrument will rest against this when sightings are made.

5 Now you have to make sure that you set up the plane table so that point x (on the table) is directly above X (on the ground), and the table is level.

You can ensure that x is directly above X by using a plumbing fork, but for most school activities it is sufficient to estimate the position by eye.

6 Level the plane table with a spirit level.

7 Use your sighting device to draw light lines from x out to the features that need to be mapped.

8 Name each feature with a different letter.

9 Measure the distance to the furthermost feature and use this distance to decide on an appropriate scale for your map.

X

Plumbingfork

PlanePlottingx

tablepaper

Investigation Plane table surveying using radiation

64 NEW QMATHS 12A

Intersection methodThe intersection method is used when all necessary features can be viewed from two stations and when features are not able to be chained.

Consider the situation where the boundary of a forest has to be surveyed but it is not possible to chain along it.

The survey would be conducted in the following way.

1 Start by selecting a suitable base line, AB, such that all points P, Q, R, S and T are visible from either end of the base line.

2 Measure the base line carefully. Check the measurement of AB to ensure that it is accurate.

10 Indicate the position of the feature on the map with a lower-case (small) letter.

11 Measure the distances to all other features, convert the distances using the scale you have chosen and indicate their positions on the map.

12 The area surveyed can be shown on the map by joining all the positions with straight lines.

pq

rs

x

P

Q

R

S

P

Q R

S

T

A B

Forest

Base line


3 Set up the plane table directly over A and level it. Mark point a on the paper on the plane table directly above A. Insert a pin at a to rest the alidade against when sighting.

4 Now use the sighting instrument to take sightings firstly to B and then to P, Q, R, S and T on the boundary of the forest. For each sighting, draw a faint ray on the paper on the plane table.

5 Choose a suitable scale and draw ab using that scale.

6 Set up the plane table directly above B. Follow the same procedure at B as at A.

7 Find the positions of P, Q, R, S and T at the intersections of the sighting lines drawn from a and b. These points are indicated by p, q, r, s and t on the map.

Traverse using the plane tableA traverse can be completed using a plane table. The automatic angle-finding feature of the plane table is used to map the features that are visible at each station. The four-sided region PQRS can be traversed using a plane table in the following way.

1 Locate the features at P, Q, R and S.

2 Set the plane table up directly above P and level it. Point p is then marked on the paper and a pin inserted at this point.

P

Q R

S

T

a b

p

q r st

ba

p q p q

r

P Q

S R

s

To R

p

To S

r

q

66 NEW QMATHS 12A

3 Take sightings to stations Q and S with the alidade against the pin at p. Draw in rays to Q and S faintly. Also take a sighting to R and draw it as a dashed line.

4 Chain PQ and use a suitable scale to plot it on the paper as pq.

5 Set up the table directly above Q and level it.

6 Orient the table so that pq on the plan is directly in line with PQ. Do this by sighting back towards P from Q.

7 Take a sighting to R and draw in a faint ray. This ray will intersect the dashed line drawn from p at r. Chain the distance QR and use the same scale used previously to draw qr on the plan. The scaled position of r should coincide with the position of r already located.

8 Use a similar procedure at stations R and S.

If you have completed the traverse accurately, you will finish up where you began.

Modelling and problem solving1 The results of a radial plane table survey of a children’s play area are recorded as follows.

Make a drawing of the play area using a scale of 1 : 1000.

2 A plane table survey of a small field is conducted and the results are shown here.

a How long is the survey line AB?b Find the bearings of AQ, BP and BR.c What are the distances AR, RB and BQ?

Feature Bearing from X Distance (m) from X

Tree 061° 83

Bench 097° 59

Hut 194° 22

Swings 273° 24

Barbecue 328° 27

N

P

Q

R

A B

Scale 1 : 2500

Exercise 2.6 Plane table surveying


3 When the region ABCD is surveyed using a plane table, the following plot results.

The survey line XY measures 90 m.a Calculate the scale.b Calculate the perimeter of the region.c Find the bearings of XB, XD, YC and YA.

4 Part of a plane table traverse of a four-sided piece of farmland is shown below.

The final corner of the region is located at X, which is at a bearing of 309° from U and 239° from W.a If UV measures 228 m, what scale has been used?b Complete a scale drawing of the survey using the same scale as the partially

completed sketch shown here.c Find the perimeter of the region.

N

A

B

C

D

YX

N

011°

149°

U

V

W

191°

Chapter summary

2

68 NEW QMATHS 12A

Chapter Review

Communication and justification1 What is meant by the term ‘booking a survey’?

2 What is the purpose of a datum point?

3 What unit of measurement would be suitable to express the area of:

a a school building?

b the school grounds?

4 What is involved in the surveying technique called triangulation?

5 Describe the method of completing an open traverse.

6 Several groups of students complete an open traverse using the same stations. When the bearings and back bearings are checked, it is found that each group finds a similar discrepancy on the same leg of the traverse. What is the most likely cause of this discrepancy? Give a reason for your answer.

7 Under what circumstances would it be suitable to use the radiation survey method with a plane table?

Knowledge and procedures8 Hau takes 37 paces to cover 30 m. Calculate the following distances he paced out.

a 25 paces along the school cricket pitch

b 277 paces from his house to the bus stop

9 Nora consistently covers 40 m in 46 paces. Calculate the area of each of the following figures paced out by Nora.

10 Complete the table by using a protractor to measure angles in the diagram at the top of the next page.

Leg Bearing Back bearing

OA

OB

OC

OD

OE

Ex 2.1

Ex 2.1

Ex 2.2

Ex 2.4

Ex 2.5

Ex 2.5

Ex 2.6

Ex 2.1

Ex 2.1

a b

87 paces38 pac

es

118 paces

65 paces

46 paces

59 paces

Ex 2.1


11 What is the back bearing for each of the following?

a 203º b 111º c 332º d 023º

12 Levels of 5.4853 m and 1.1072 m are taken from two points on a slope separated by a horizontal distance of 46.1022 m.

a What is the fall between the two points?

b At what angle does the land slope with the horizontal?

c Express the fall as a ratio in the form 1 : x.

13 What fall is required for a storm water drain that runs 40.5 m down one side of a house if a fall of 1 : 200 is needed to ensure that water drains sufficiently?

14 In each of the following regions, all measurements are in metres. Calculate the area of each region.

N

A

B

C

D

E

O

Ex 2.1

Ex 2.1

Ex 2.1

Ex 2.2

a b

82

30

33 54

58

46

80.452.3

48.961.2

99.3

70 NEW QMATHS 12A

Modelling and problem solving15 The survey notes below relate to the survey of a play area at a school. The area was

paced out by Robbie, who covers 50 m every 56 paces. (All dimensions are in paces.)

a Make a scale drawing of the area using a scale of 1 : 2000.b What is the area of this region?

16 Two irregularly shaped regions are surveyed and the results recorded as follows. (All dimensions are in metres.)

In each case, do the following.i Select a suitable scale and make a scale drawing of the area.

ii Calculate the area of the irregular region surveyed.

17 A river obstructs a main survey line as shown at right. The surveyor finds that:

AO = 45 mPO = 117 m

Find the length of AB.

Ex 2.1

Ex 2.2

a b

45 m

117 m

A B

P Q

RiverO

Ex 2.3


18 A field with irregular boundaries is surveyed. The field notes from the survey are as follows. (All dimensions are in metres.)

a Use a suitable scale to make a scale drawing of the field.b Use the scale drawing that you have made to estimate the area of the field.

19 A picnic ground near the sea is surveyed. Use the field notes supplied to make a scale drawing of the area. (All dimensions are in metres.)

20 The details of a compass traverse with XY = 100 m are as follows. Use a scale of1 : 2500 to draw a plan from these details.

Station Bearing from X Bearing from Y

A 070º 012º

B 122º 142º

C 135º 180º

D 214º 239º

E 005º 310º

F 038º 010º

Ex 2.4

Ex 2.4

Ex 2.5

72 NEW QMATHS 12A

21 By choosing a suitable scale, plot a map to show the traverse for which the details are shown at right. Mark all relevant features on the map. (Dimensions are in metres.)

22 The diagram at right is a scale drawing of a proposed recreation area. The position of the main survey line AB is shown.a Use the drawing to complete

the table below.b Calculate the scale and the

perimeter of the recreation area.

23 A surveyor completed a survey of a field (PQRS). The field notes are shown as follows. AB is the main survey line. Make a scale drawing of the field using a suitable scale.

Station Bearing from A

Bearing from B

P

Q

R

S

T

Ex 2.5

Ex 2.6

N

P

S

T

R

Q

A B90 m

Recreation area

Ex 2.6

Documents

Surveys and site plansmathsbooks.net/New Q Maths 12A/ch02 Surveys and site plans.pdf · SURVEYS AND SITE PLANS 37 CHAPTER 2 3 Wendel has a steady pace and knows that he takes 80 paces