5. CALCULATION OF AREA IN SURVEYING

 

CALCULATION OF AREA IN SURVEYING

(Civil Engineering)

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1. Introduction

In surveying, calculation of area means finding the surface area of a piece of land or plot from field measurements.
It is essential for:

• Land records and property valuation

• Estimation of earthwork

• Planning of buildings, roads, canals

• Agricultural land measurement

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2. Types of Areas in Surveying

1. Regular areas – square, rectangle, triangle, circle

2. Irregular areas – fields, ponds, uneven boundaries

3. Areas from maps – measured using plan or drawing

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3. Methods of Calculating Area

A. Area of Regular Figures

B. Area of Irregular Figures

C. Area by Coordinates

D. Area from Plan / Map

E. Area by Offsets

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4. Calculation of Area of Regular Figures

(a) Square

$$Area = a^2$$

(b) Rectangle

$$Area = L \times B$$

(c) Triangle

$$Area = \frac{1}{2} \times base \times height$$

(d) Circle

$$Area = \pi r^2$$

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Example 1 – Rectangular Plot

Length = 40 m, Breadth = 25 m

$$Area = 40 \times 25 = 1000 \, m^2$$

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5. Area of Irregular Figures

Irregular boundaries are common in field surveys. These areas are found using:

1. Mid-ordinate rule

2. Average ordinate rule

3. Trapezoidal rule

4. Simpson’s rule

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5.1 Mid-Ordinate Rule

Offsets are taken at equal intervals perpendicular to a baseline.

$$Area = d \times \sum M$$

Where:

• $d$ = common interval

• $M$ = mid-ordinates

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Example 2 – Mid-Ordinate Rule

Mid-ordinates (m)	2.0	2.5	3.0	2.8	2.2

Spacing $d = 10 m$

$$Area = 10 \times (2 + 2.5 + 3 + 2.8 + 2.2)$$ $$Area = 10 \times 12.5 = 125 \, m^2$$

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5.2 Average Ordinate Rule

$$Area = L \times \frac{\sum O}{n}$$

Where:

• $L$ = length of baseline

• $O$ = ordinates

• $n$ = number of ordinates

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5.3 Trapezoidal Rule

$$Area = d \left[\frac{O_1 + O_n}{2} + O_2 + O_3 + \dots + O_{n-1}\right]$$

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Example 3 – Trapezoidal Rule

Offsets at 10 m intervals:

| Offset (m) | 1.5 | 2.0 | 2.8 | 2.2 | 1.6 |

$$Area = 10 \left[\frac{1.5 + 1.6}{2} + 2.0 + 2.8 + 2.2\right]$$ $$Area = 10 (1.55 + 7.0) = 85.5 \, m^2$$

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5.4 Simpson’s Rule

(Most accurate for curved boundaries)

$$Area = \frac{d}{3} \left[(O_1 + O_n) + 4(O_2 + O_4 + \dots) + 2(O_3 + O_5 + \dots)\right]$$

Note: Number of intervals must be even.

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6. Area by Coordinates (DMD Method)

When the coordinates of the boundary points are known, area is calculated using:

$$Area = \frac{1}{2} \sum (x_1y_2 - x_2y_1)$$

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Example 4 – Coordinate Method

Point	x (m)	y (m)
A	10	10
B	40	10
C	40	30
D	10	30

$$Area = \frac{1}{2}[(10×10 + 40×30 + 40×30 + 10×10) − (10×40 + 10×40 + 30×10 + 30×10)]$$ $$Area = 600 \, m^2$$

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7. Area from Plan (Using Scale)

If a plot is drawn on a map:

$$Ground \, Area = (Map \, Area) \times (Scale)^2$$

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Example 5 – Area from Map

Map area = 20 cm²
Scale = 1 : 1000

$$Ground \, Area = 20 \times (1000)^2 = 20,000,000 \, cm^2$$ $$= 2000 \, m^2$$

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8. Area by Planimeter

A planimeter is a mechanical/digital instrument used to measure irregular areas directly from a map.

Used in:

• Town planning

• Forest area measurement

• Lake and reservoir areas

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9. Sources of Error in Area Calculation

1. Incorrect offsets

2. Unequal spacing

3. Wrong plotting

4. Instrument errors

5. Reading mistakes

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10. Precautions

• Offsets must be taken perpendicular

• Use Simpson’s rule for curved boundaries

• Check arithmetic twice

• Use correct scale

• Keep field notes neat

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11. Applications in Civil Engineering

• Land acquisition

• Building layout

• Road widening projects

• Irrigation planning

• Real estate development

• Tax assessment

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12. Comparison of Methods

Method	Accuracy	Use
Mid-ordinate	Low	Rough estimates
Trapezoidal	Medium	Field work
Simpson’s	High	Curved boundaries
Coordinate	Very high	GPS / Total station
Planimeter	High	Maps & plans