Contents

# Introduction

Many surveys have the primary goal of finding the area of the surveyed tract and the amount of earthwork.

## Measurement of Area

• The method adopted for measurement of area and volume depends upon
1. Accuracy required
2. Shape/ Geometry of track.
• Measurement can be done by either
1. By field measurement
2. By plan measurement
• If the plan is enclosed by a straight line, it can be divided into geometrical figures, like: triangle, rectangle, square etc. The area of these figures can be determined by using standard formulas.
• But if the boundaries are irregular, then approximate methods are being used.

Note: Planimeter meter is used to determine areas of irregular shape.

## Computation of Area of Geometrical Figures

### 7.Trapezium

Area (ABCD) = Area(ABC) + Area(ACD)

### 8.Oblique Triangle

$$A=\sqrt{s(s-a)(s-b)(s-c)}$$

where a, b, care the sides.

$$s=\frac{a+b+c}{2}$$

### 9.Area of a Segment

Area of segment (ACB) = Area of sector (ACBO) – Area of triangle (AOB)

$$A=R^{2}\left ( \frac{\Pi \alpha }{360^{\circ}} -\frac{sin}{2}\right )$$

### 10.Regular Polygon

Area= length of perimeter ×  half of the perpendicular distance from the centre of sides.

$$A=\frac{nL^{2}}{4}cot^{2}\frac{180^{\circ}}{n}$$

where n is the number of sides and L the length of one sides

## Computation of Area of Irregular Shape

### 1.Mid Ordinate Rule

If offsets h1, h2, …., hn-1. are measured at the mid point of each division.

Area = average ordinate × length of base.

$$A=\frac{h_1+h_2+….+h_{n-1}}{n-1}L$$.

∴L=(n-1)d

where

n-1 = number of division

d = distance between the two perpendicular offsets.

L = length of base line {L=(n-1)d}

### 2.Average Offset Rule

If offsets O1, O2, …., On. are measured at the mid point of each division and spaced apart at equal distance d.

Area = average of all ordinate × length of base.

$$A=\frac{O_1+O_2+….+O_n}{n}L$$

$$A=\frac{(n-1)d}{n}\Sigma O_i$$

### 3.Trapezoidal Rule

The accuracy of this method is more than mid-ordinate and average ordinate method.

Area of first trapezoid $$=\frac{O_1+O_2}{2}d$$.

Area of second trapezoid $$=\frac{O_2+O_3}{2}d$$.

Area of last trapezoid $$=\frac{O_{n-1}+O_n}{2}d$$.

Total area of trapezoid

### 4.Simpson’s One Third Rule

• If the boundaries of irregular tract are curved then Simpson method is preferred over the trapezoidal rule to calculate the area of given track.
• In this the curved boundary is considered to be a parabolic arch.

Note

• To apply Simpson’s rule offsets should be odd in number.
• To work with even no of offsets Simpson’s should be apply upto last offsets and remaining area should be calculated using trapezoidal rule.

# Measurement Of Volume

The computation of the volume of different quantities is required for planning & designing  of various engineering work .

1. Earthwork for highway, railway, retaining walls.
2. Volume for reservoir capacity
3. Concreting work
4. Storage requirement

## Computation of Volume of Irregular Shape

### 1.Trapezoidal Rule / End Area Method

This method is also called as end are method.

### 2.Prismoidal / Simpson’s  Rule

Note

• This formula usually gives less volume than trapezoidal formula.
• This formula is not suitable for rock excavation and concrete work.
• To apply Simpson’s rule offsets should be odd in number.
• To work with even no of offsets Simpson’s should be apply upto last offsets and remaining volume should be calculated using trapezoidal rule.

### 1 thought on “Measurement of Area and Volume (Mid Ordinate Rule, Average Offset Rule, Trapezoidal Rule, Simpson’s Rule)”

1. Amazing detailed elaborative concepts. It is only examples you have not included

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