Comprehensive Study Guide: Engineering Surveying I - Basics of Surveying and Horizontal Distance Measurements

Course Overview & Syllabus Details

Course Code: EGCV2120
Course Title: Engineering Surveying I
Credit Hours: $3\text{ Credit Hours}$
Prerequisites: None
Department: Civil and Architectural Engineering, Department of Engineering
Institution: University of Technology and Applied Sciences, Sultanate of Oman
Course Goal: Impart knowledge about the principles of surveying, the use of a range of survey equipment in the field to measure, and the preparation of required data for civil engineering works.

Course Overview / Chapters

Chapter 1: Basics of Surveying and Horizontal Measurement
Chapter 2: Levelling
Chapter 3: Angles and Directions
Chapter 4: Measurement of Areas and Volumes
Chapter 5: Theodolite Surveying

Course Learning Objectives and Mapping Outcomes

Course Objective 1: Impart fundamental concepts about surveying, its applications, horizontal distance measurement, and its methods.
- Mapped Outcome 1: Gain knowledge about basics of surveying, classification, and types of surveying; introduction to maps and plans with different scaling.
Course Objective 2: Illustrate different methods for measuring angles and directions.
- Mapped Outcome 2: Distinguish between the concepts of meridians and bearings, measuring horizontal and vertical angles by angle-measuring instruments.
Course Objective 3: Demonstrate knowledge about the determination of elevations and its methods.
- Mapped Outcome 3: Analyze basic concepts of elevation and its methods to find the difference in level between any two points.
Course Objective 4: Examine concepts of computation of area and volume.
- Mapped Outcome 4: Identify different methods for calculating area and volume.
Course Objective 5: Acquire knowledge about the basics of curves and their applications in the field.
- Mapped Outcome 5: Determine distance and elevations by tachometry.
- Mapped Outcome 6: Interpret the uses of curves in surveying and their types.
Course Objective 6: Illustrate appropriate methods to measure various parameters at a construction site and their practical applications.
- Mapped Outcome 7: Execute practical work related to surveying by using basic and modern surveying instruments.

Assessment Plan & Rubric

Mid Exam:
- Weight: $20\%$
- Learning Outcomes Mapping: Outcomes 1 & 3
TEST:
- Weight: $10\%$
- Learning Outcomes Mapping: Outcome 1
FINAL EXAM:
- Weight: $40\%$
- Learning Outcomes Mapping: Outcomes 1, 2, 3, 4, 5 & 6
PRACTICAL:
- Weight: $30\%$
- Learning Outcomes Mapping: Outcomes 1, 2, 3, 4, 5 & 7
Total Weight: $100\%$

Instructions to Students

Be regular and punctual to all classes.
Carry handouts (hard or soft copy) for every class.
Bring a pen, scale, pencil, eraser, and CASIO scientific calculator.
Use of mobile phones inside the class is prohibited.
Attend classes in the specific section registered.

Basics of Surveying and Definitions

Surveying Definition: The art and science of determining the relative positions of points above, on, or beneath the surface of the earth, or establishing such points, and representing them graphically or numerically (Bannister).
Land Surveying Definition: The detailed measurement and description of land, including its boundaries, topography, and dimensions.

Objectives of Surveying

To determine the relative positions of points on the earth's surface.
To set out lines and grades needed for the construction of buildings, roads, dams, and other structures.
To calculate areas, volumes, and other related quantities.
To prepare necessary maps and diagrams of an area.

Classification of Surveying

Primary Classifications

Plane Surveying:
- Assumes the earth's surface is flat (a horizontal plane).
- Neglects the curvature of the earth.
- Applicable for relatively small areas of less than $250\,\text{km}^2$ .
- Plumb lines across different points are treated as parallel lines.
- Used for most engineering construction projects such as canals, railways, highways, buildings, and pipelines.
Geodetic Surveying:
- Takes into account the true spheroid shape and curvature of the earth's surface.
- Applicable for large survey areas exceeding $250\,\text{km}^2$ .
- Plumb lines converge toward the center of the earth.
- Carried out for highly precise works.
- Crucial for national mapping, land development, mining, agriculture, and environmental management.

Branches classified by Purpose

Control Surveying: Establishes horizontal and vertical positions of control points used as reference points for other surveys.
Land Surveying (Planimetric / Property / Boundary / Cadastral Survey): Determines the boundaries and areas of land parcels.
Topographic Survey: Prepares a plan or map of a region showing natural and man-made features along with structural elevations.
Engineering Survey: Collects requisite data for planning, design, and execution of engineering projects. Divided into three steps:
1. Reconnaissance Survey: Explores site conditions and availability of infrastructures.
2. Preliminary Survey: Collects adequate data to prepare a plan/map of the area for design purposes.
3. Location Survey: Sets out work on the ground for actual construction and execution.
Route Survey: Plans, designs, and lays out routes for linear projects including highways, railways, canals, and pipelines.
Construction Survey: Establishes points, lines, and grades required for staking out engineering works after designs are finalized.
Astronomic Survey: Determines latitude, longitude, and azimuth of observation stations using astronomical observations.
Mine Survey: Special survey procedures conducted specifically for surface (opencast) and underground mining operations.

Branches classified by Nature of Field

Land Surveying
Marine (Hydrographic) Surveying
Astronomical Surveying

Branches classified by Survey Methods

Triangulation Surveying: Survey stations are connected with survey lines forming a network of triangles. Used for large areas with uneven boundaries. Triangle formulas are applied to solve dimensions.
Traverse Surveying: The survey area is divided into a series of connected lines forming an open or closed polygon or framework.

Principles of Surveying

Principle 1: Work from the Whole to the Part
- The entire area is first enclosed by main controlling stations and main survey lines.
- The area is subsequently divided into smaller parts by forming well-conditioned triangles.
- Main lines are measured with high accuracy to prevent the accumulation of errors. Minor details are filled in later. If work progresses from part to whole, small errors multiply and accumulate over large areas.
Principle 2: Locate New Stations by at Least Two Measurements from Fixed Reference Points
- Positions of new stations must always be anchored using a minimum of two reference measurements from established reference points.
- Measurements can be linear (using tape/chain) or angular (using theodolite/compass), or a combination of both.

Plans and Maps

Plan vs Map Comparison

Definitions & Comparative Features

Plan:
- Graphical representation to scale of features on, near, or below the surface of the earth as projected onto a horizontal plane.
- Drawn on a large scale (e.g., $1\,\text{cm} = 10\,\text{m}$ or scale $< 10\,\text{m}$ , such as $1\,\text{cm} = 50\,\text{m}$ or $1:500$ ).
- Covers a small area with high detail (exact dimensions of buildings, road widths, structural elements).
- Depicts horizontal distances and directions.
Map:
- Visual or graphical representation of a large portion of the earth's surface on a flat sheet of paper on a small scale.
- Drawn on a small scale (e.g., $1\,\text{cm} = 100\,\text{m}$ or scale $> 100\,\text{m}$ , such as $1\,\text{cm} = 50\,\text{km}$ or $1:500000$ ).
- Covers a large area showing key relationships (distance, direction, size) between major features without exact structural detail.
- Topographic maps explicitly depict vertical distances (elevations) using contours.

Types of Maps

Physical (Relief) Map: Displays natural geographical features of the earth such as mountains, plateaus, plains, rivers, and oceans.
Thematic Map: Displays specific themes or data distributions such as climate, rainfall, vegetation, land use, population density, or life expectancy.
Political Map: Displays administrative and political boundaries of cities, towns, governorates, states, and national borders.
Topographic Map: Utilizes contour lines to show relief, shape, and elevation of terrain. Closely spaced contour lines represent steep terrain, whereas widely spaced lines represent flat terrain.

Types of Plans

Architectural Plans: Includes site plans, floor plans, and sectional drawings detailing geometry and layout.
Structural Plans: Details structural engineering elements including column layouts, footing designs, beam schedules, and reinforced concrete slab reinforcement.

Fundamental Concepts & Computations

Rounding Off Rules

Rule 1: If the digit following the rounding digit is $1$ , $2$ , $3$ , or $4$ , drop all digits to the right of the rounding digit.
- Example 1.1: Round off $3.423$
  - To nearest Hundredths: $3.42$
  - To nearest Tenths: $3.4$
  - To nearest Units: $3$
Rule 2: If the digit following the rounding digit is $5$ , $6$ , $7$ , $8$ , or $9$ , add $1$ to the rounding digit and drop all digits to the right.
- Example 1.2: Round off $2.856$
  - To nearest Hundredths: $2.86$
  - To nearest Tenths: $2.9$
  - To nearest Units: $3$
- Example 1.3: Round off $2.315$ and $2.325$ to the nearest hundredths:
  - $2.315 \rightarrow 2.32$
  - $2.325 \rightarrow 2.33$

Units of Measurement

Linear & Area Conversions

$1\,\text{ha} = 10000\,\text{m}^2$
$1\,\text{acre} = 4046.86\,\text{m}^2$
$1\,\text{m}^2 = 0.0001\,\text{ha}$
$1\,\text{m}^2 = 0.000247105\,\text{acre}$

Worked Example 1.4

Convert $9250\,\text{m}^2$ to:

Hectares: $9250\,\text{m}^2 = \frac{9250}{10000} = 0.925\,\text{ha}$
Acres: $9250\,\text{m}^2 = \frac{9250}{4046.86} = 2.286\,\text{Acre}$

Angular Units

Measured in degrees ( $^\circ$ ), minutes ( $'$ ), and seconds ( $''$ ).
Sexagesimal System:
- $1^\circ = 60'$
- $1' = 60''$
Conversion Examples:
- Express $85.145^\circ$ in angular sexagesimal units:
  - Whole degrees = $85^\circ$
  - $0.145 \times 60' = 8.7'$ (Whole minutes = $8'$ )
  - $0.7 \times 60'' = 42''$
  - Result: $85^\circ 8' 42''$
- Express $45^\circ 32' 47''$ in decimal degrees:
  - $\text{Decimal Degrees} = 45 + \frac{32}{60} + \frac{47}{3600} = 45.546388...^\circ \approx 45.546^\circ$
- Degree to Radian Conversion: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Scaling

Definition: Scale is the ratio of the distance on the drawing/map to the corresponding actual horizontal distance on the ground.
Types of Scale:
- Full-Size Scale: Actual length is represented in full size ( $1:1$ ).
- Reducing Scale: Actual length is reduced to fit the drawing sheet.
- Increasing / Enlarging Scale: Small actual features are enlarged to display fine details clearly.

Scale Classification Ranges

For Maps:
- Large scales: $< 1:2000$
- Intermediate scales: $1:2000$ to $1:10000$
- Small scales: $1:10000$ to $1:100000000$
For Plans:
- Site plans: $1:50$ to $1:500$
- Detail plans: $1:1$ to $1:20$

Methods of Expressing Scale

Verbal Scale (In Words): Stated directly in words, e.g., "One inch represents 16 miles".
Graphical / Linear Scale: Drawn directly on the map as a graduated bar/ruler line representing ground distances. Remains accurate even if paper shrinks or expands.
Representative Fraction (RF Scale): Expressed as a unitless ratio or fraction, e.g., $1/24000$ or $1:24000$ , where $1$ unit on paper equals $24000$ of the same units on the ground.

Scale Ordering Problems

Rule: A larger scale has a smaller denominator fraction value. Smaller representative fractions indicate smaller scales.

Example 1.6

Arrange scales from Smallest to Largest: Given: $1\,\text{m}:10000\,\text{m}$ , $1\,\text{m}:1000\,\text{m}$ , $1\,\text{m}:100\,\text{m}$ , $1\,\text{m}:10\,\text{m}$

Smallest to Largest:
1. $1\,\text{m}:10000\,\text{m}$
2. $1\,\text{m}:1000\,\text{m}$
3. $1\,\text{m}:100\,\text{m}$
4. $1\,\text{m}:10\,\text{m}$

Practice Problem 1.5

Arrange the following scales from Smallest to Largest:

Option A: $1\,\text{cm} : 50\,\text{km} \rightarrow 1\,\text{mm} = 5\,\text{km}$
Option B: $1\,\text{mm} : 1000\,\text{m} \rightarrow 1\,\text{mm} = 1\,\text{km}$
Option C: $1\,\text{m} : 50000\,\text{km} \rightarrow 1000\,\text{mm} = 50000\,\text{km} \rightarrow 1\,\text{mm} = 50\,\text{km}$
Option D: $1\,\text{mm} : 500\,\text{m} \rightarrow 1\,\text{mm} = 0.5\,\text{km}$

Order from Smallest to Largest:

$1\,\text{m} : 50000\,\text{km}$ ( $1\,\text{mm} = 50\,\text{km}$ )
$1\,\text{cm} : 50\,\text{km}$ ( $1\,\text{mm} = 5\,\text{km}$ )
$1\,\text{mm} : 1000\,\text{m}$ ( $1\,\text{mm} = 1\,\text{km}$ )
$1\,\text{mm} : 500\,\text{m}$ ( $1\,\text{mm} = 0.5\,\text{km}$ )

Map Scale Measurement Problem

Calculate distance $AB$ in kilometers given map scale $1\,\text{m} : 750\,\text{km}$ and measured paper length $9.5\,\text{cm}$ :

Scale conversion: $100\,\text{cm} = 750\,\text{km} \Rightarrow 1\,\text{cm} = 7.5\,\text{km}$
Distance $AB$ : $\text{Distance } AB = 9.5\,\text{cm} \times 7.5\,\text{km/cm} = 71.25\,\text{km}$

Horizontal Distance Measurement Methods

Basic Principles

Two points form a line.
The shortest distance between two points is a straight line.

Surveying Methods for Distance Measurement

Pacing: Counting natural walking steps. Error is roughly $2\%$ $\text{Distance } (D) = \text{No. of paces} \times \text{Pace Factor } (PF)$
Odometer: A wheel rolled along the ground that converts rotations to distance.
Chaining / Taping: Direct measurement using steel tapes or chains.
Stadia (Tachometry): Optical method using theodolite and leveling rod.
Electronic Distance Measurement (EDM): Uses phase shift or pulse timing of electromagnetic/laser waves.
Global Positioning System (GPS): Satellite-based signal positioning.

Taping Equipment & Accessories

Accessories used in Taping

Engineer's Steel Tape: Graduated tape for linear measurement.
Hand Level: Used to maintain tape horizontally on sloping ground.
Marking Pins (Arrows): Steel pins to mark full tape-length intervals.
Range Pole (Ranging Rod): Wood/metal pole painted red/white to mark line alignments.
Tension Handle / Spring Balance: Standardizes applied pull/tension.
Clamping Handle: Clamps tape securely without damaging it.
Plumb Bob: Transfers point vertically from suspended tape level to ground peg.

Taping Procedure Steps

Aligning the tape along the straight line.
Stretching the tape under standard tension.
Plumbing elevated ends over ground marks.
Marking and tallying full tape lengths.
Measuring remaining fractional tape lengths.

Basic Surveying Terms

Survey Framework and Lines

Survey Stations: Fixed control points on the ground defining survey lines.
- Main Stations: Primary boundary corner points enclosing the survey.
- Tie / Subsidiary Stations: Points selected along main lines to run secondary/tie lines.
Main Survey Lines: Lines joining main survey stations.
Base Line: The longest main survey line that forms the core backbone framework of the survey layout.
Check Line: A line run from the apex of a triangle to a fixed point on the opposite base line to verify geometric accuracy.
Tie Line: A secondary line connecting tie stations to capture interior site features and details.
Offsets: Lateral perpendicular or oblique distance measurements from features/objects to the chain line.

Perpendicular and Oblique Offsets

Perpendicular Offsets: Lateral measurements taken at an exact $90^\circ$ right angle to the survey line.
Oblique Offsets: Lateral measurements taken at an angle other than $90^\circ$ when direct perpendicular layout is obstructed.

Errors in Surveying & Tape Corrections

Error Definition: Difference between the measured value and the true/exact value of a quantity.

Systematic Taping Corrections

1. Correction for Absolute Length / Standard Length ( $C_l$ )

When a tape's actual length $L'$ differs from its nominal standard length $L$ :

Correction per tape length $c = L' - L$
Total correction $C_l = \frac{c}{L} \times D$
$\text{Corrected Length } D' = D \times \left(\frac{L'}{L}\right)$
Rule: If tape is too short ( $L' < L$ ), measured distances read higher than actual, so correction $C_l$ is negative ( $-$ ). If tape is too long ( $L' > L$ ), correction $C_l$ is positive ( $+$ ).

2. Temperature Correction ( $C_t$ )

$C_t = \alpha \times (T_m - T_o) \times L$

$\alpha$ = Coefficient of thermal expansion ( $\text{per } ^\circ\text{C}$ )
$T_m$ = Temperature during field measurement ( $^\circ\text{C}$ )
$T_o$ = Standard calibration temperature ( $^\circ\text{C}$ )
$L$ = Measured length ( $\text{m}$ )

3. Pull / Tension Correction ( $C_p$ )

$C_p = \frac{(P_m - P_o) \times L}{A \times E}$

$P_m$ = Applied pull during field measurement ( $\text{N}$ )
$P_o$ = Standard calibration pull ( $\text{N}$ )
$A$ = Cross-sectional area of tape ( $\text{mm}^2$ )
$E$ = Modulus of Elasticity ( $\text{N/mm}^2$ )
$L$ = Measured length ( $\text{m}$ )

4. Sag Correction ( $C_s$ )

Applied when tape is suspended at support points without continuous ground contact:

$C_s = - \frac{W^2 \times L}{24 \times P_m^2} = - \frac{w^2 \times L^3}{24 \times P_m^2}$

$W$ = Total weight of suspended tape span ( $\text{N}$ )
$w$ = Weight of tape per unit length ( $\text{N/m}$ )
$P_m$ = Applied pull ( $\text{N}$ )
Note: Sag correction is ALWAYS negative ( $-$ ).

5. Slope Correction ( $C_h$ )

$C_h = - \frac{h^2}{2L} \quad \text{or} \quad C_h = - L(1 - \cos\theta)$

$h$ = Elevation difference between tape ends
$\theta$ = Slope angle
Note: Slope correction is ALWAYS negative ( $-$ ).

Exercise Problems on Tape Corrections

Exercise Problem 1

A rectangular plot was measured using a $50\,\text{m}$ steel tape which was found to be $0.025\,\text{m}$ too short. Recorded length = $180.455\,\text{m}$ , recorded width = $127.062\,\text{m}$ . Calculate actual/correct area of the plot.

Solution Steps:
- Nominal length $L = 50\,\text{m}$
- Actual tape length $L' = 50 - 0.025 = 49.975\,\text{m}$
- $\text{Actual Length} = \left(\frac{49.975}{50}\right) \times 180.455 = 180.3647725\,\text{m}$
- $\text{Actual Width} = \left(\frac{49.975}{50}\right) \times 127.062 = 126.998469\,\text{m}$
- $\text{Actual Area} = 180.3647725\,\text{m} \times 126.998469\,\text{m} = 22906.04\,\text{m}^2$

Exercise Problem 2

A distance of $582.35\,\text{m}$ was measured with a $30\,\text{m}$ steel tape standardized under $T_o = 20^\circ\text{C}$ and $P_o = 50\,\text{N}$ . Tape standard length $L' = 30.0052\,\text{m}$ . Field conditions: $T_m = 32^\circ\text{C}$ , $P_m = 100\,\text{N}$ . Tape suspended at ends, total weight $W = 8\,\text{N}$ . Tape properties: $\alpha = 1.15 \times 10^{-5}\,/^\circ\text{C}$ , $A = 2\,\text{mm}^2$ , $E = 2 \times 10^5\,\text{N/mm}^2$ .

Individual Corrections Summary:
- Absolute length correction $C_l = +0.101\,\text{m}$
- Combined temperature & pull correction $C_{t+p} = +0.0097\,\text{m}$
- Sag correction $C_s = -0.99\,\text{m}$
- $\text{Total Correction} = +0.101 + 0.0097 - 0.99 = -0.8987\,\text{m}$
- $\text{Corrected Distance} = 582.35 - 0.8987 = 581.45\,\text{m}$

Exercise Problem 3

A single span distance of $20\,\text{m}$ is measured using a $30\,\text{m}$ steel tape supported at ends under pull $P_m = 80\,\text{N}$ ( $P_o = 50\,\text{N}$ ), $T_m = 25^\circ\text{C}$ ( $T_o = 20^\circ\text{C}$ ), unit weight $w = 0.25\,\text{N/m}$ , $A = 3\,\text{mm}^2$ , $E = 2 \times 10^5\,\text{N/mm}^2$ , $\alpha = 1.15 \times 10^{-5}\,/^\circ\text{C}$ .

Individual Corrections:
- Temperature correction $C_t = +0.00113\,\text{m}$
- Pull correction $C_p = +0.0031\,\text{m}$
- Sag correction $C_s = -0.002128\,\text{m}$
- $\text{Total Correction} = +0.0031 + 0.00113 - 0.002128 = 0.002102\,\text{m}$
- $\text{Corrected Distance} = 20 + 0.002102 = 20.002102\,\text{m}$

Obstacles in Linear Measurement

An obstacle is an object (pond, building, river, jungle) intersecting a survey line that prevents direct linear taping along that line.

Categories of Obstacles

Obstacle to Chaining but not Vision: (e.g., rivers, ponds, sheets of water).
Obstacle to Vision but not Chaining: (e.g., trees, jungles, small hills).
Obstacle to both Chaining and Vision: (e.g., tall buildings, large structures).

Worked Numerical Examples for Obstacles

1. River Obstacle via Similar Triangles

River Obstacle Survey Geometry

A chain line crosses a river at $90^\circ$ . Given $AB = 57.73\,\text{m}$ , $BD = 100\,\text{m}$ , and $\angle ABD = \angle ADC = 90^\circ$ . Calculate width $BC$

Geometric Relation: By similar right triangles ( $\triangle ABD \sim \triangle DBC$ ): $\frac{AB}{DB} = \frac{DB}{BC} \Rightarrow BC = \frac{DB \times DB}{AB}$
Calculation: $BC = \frac{100 \times 100}{57.73} = \frac{10000}{57.73} = 173.22\,\text{m}$

2. Building Obstacle via Right-Angle Trigonometry

Building Obstacle Geometry

A building obstructs survey line $DE$ . Line $BC = 200\,\text{m}$ is offset perpendicular to line $AB$ . Given $\angle BCD = 45^\circ$ and $\angle BCE = 60^\circ$ . Calculate lengths $CD$ and $CE$

Calculation for $CD$ : $\cos(45^\circ) = \frac{BC}{CD} \Rightarrow CD = \frac{200}{\cos(45^\circ)} = \frac{200}{0.7071} = 282.84\,\text{m}$
Calculation for $CE$ : $\cos(60^\circ) = \frac{BC}{CE} \Rightarrow CE = \frac{200}{\cos(60^\circ)} = \frac{200}{0.5} = 400\,\text{m}$

3. River Width via Sine Rule

Given line $AB = 100\,\text{m}$ along near bank. Observed angles to point $C$ across river: $\angle BAC = 90^\circ - 40^\circ = 50^\circ$ $\angle ABC = 90^\circ - 25^\circ = 65^\circ$

Triangle Angles: $\angle ACB = 180^\circ - 50^\circ - 65^\circ = 65^\circ$
Sine Rule in $\triangle ABC$ : $\frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(65^\circ)} \Rightarrow \frac{100}{\sin(65^\circ)} = \frac{AC}{\sin(65^\circ)} \Rightarrow AC = 100\,\text{m}$
Width of River $w$ (perpendicular distance $AX$ where $\angle AXC = 90^\circ$ ): $\frac{AC}{\sin(90^\circ)} = \frac{w}{\sin(50^\circ)} \Rightarrow w = 100 \times \sin(50^\circ) = 76.60\,\text{m}$

4. Pond / Lake Obstacle via Cosine Rule

A pond obstructs chain line $ab$ . Line $mn = 500 + 550 = 1050\,\text{m}$ . Given $ma = 900\,\text{m}$ , $an = 1100\,\text{m}$ , $mb = 500\,\text{m}$ , $bn = 550\,\text{m}$ . Line $ab$ is not perpendicular to $mn$ . Calculate distance $ab$

Step 1: Apply Cosine Rule in $\triangle mna$ to find $\cos(\angle amn)$ : $na^2 = ma^2 + mn^2 - 2 \times ma \times mn \times \cos(\angle amn)$ $1100^2 = 900^2 + 1050^2 - 2 \times 900 \times 1050 \times \cos(\angle amn)$ $1210000 = 810000 + 1102500 - 1890000 \times \cos(\angle amn)$ $\cos(\angle amn) = \frac{-1210000 + 810000 + 1102500}{1890000} = \frac{702500}{1890000} = 0.371693 \approx 0.372$ $\angle amn = 68.17^\circ = 68^\circ 10' 47.73''$
Step 2: Apply Cosine Rule in $\triangle mab$ to solve $ab$ : $ab^2 = ma^2 + mb^2 - 2 \times ma \times mb \times \cos(\angle amb)$ $ab^2 = 900^2 + 500^2 - (2 \times 900 \times 500 \times 0.371693)$ $ab^2 = 810000 + 250000 - 334523.55 = 725476.45$ $ab = \sqrt{725331.45} = 851.66\,\text{m}$

Academic References

Bannister, A., Raymond, S. & Baker, R., Engineering Surveying, Longman Scientific and Technical, England.
Basak, N.N., Surveying and Levelling, Tata McGraw Hill Education Private Limited.
Punmia, B.C., Jain, A.K. & Jain, A.K., Surveying Vol. I & II, Laxmi Publications Pvt Ltd, New Delhi, 2005.
Nathanson, J., Surveying Fundamentals and Practices, Prentice Hall Publishers, 2011.
Irvine, W. & Maclennan, F., Surveying for Construction, Tata McGraw Hill Education Private Limited.
Pettet, J., Site Surveying and Levelling, London: Hutchinson.
Roy, S.K., Fundamentals of Surveying, Second Edition, Prentice Hall of India, 2004.
Clark, D., Plane and Geodetic Surveying Vol. I and II, Constable and Company, London, 1980.
Davis, R.E., Foote, F.S. & Kelly, J.W., Surveying: Theory and Practice, 7th Ed., McGraw-Hill, New York, 1980.