** 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**

| Area= length(L) × width(b) |

| Area= length(L) × width(b) |

| Area= half of the base(b) × perpendicular height(h) |

| Area = Π × (Radius)2 |

| Area= base × perpendicular height |

| Area= half of the sum of parallel sides × perpendicular height |

**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)

\(A=\frac{\Pi R^{2}}{360^{\circ}}\times \alpha -\frac{1}{2}\times 2Rsin\frac{\alpha }{2}\times Rcos\frac{\alpha }{2}\) |

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

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

**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 h_{1}, h_{2}, …., h_{n-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

\(A=\frac{h_1+h_2+….+h_{n-1}}{n-1}(n-1)d\) |

\(A=(h_1+h_2+…+h_{n-1})d\) |

**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 O_{1}, O_{2}, …., O_{n}. are measured at the mid point of each division and spaced apart at equal distance d.

Area = average of all ordinate × length of base.

\(A=\frac{O_1+O_2+….+O_n}{n}L\)

\(A=\frac{O_1+O_2+….+O_n}{n}(n-1)d\) |

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

**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

\(A=\frac{1}{2}(O_1+O_2)d+\frac{1}{2}(O_2+O_3)d+….+\frac{1}{2}(O_{n-1}+O_n)d\) |

\(A=\frac{1}{2}d[O_1+2O_2+2O_3+2O_{n-1}+O_n]\) |

\(A=d[\frac{O_1+O_N}{2}+O_2+O_3+…O_{n-1}]\) |

**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.

\(A=\frac{d}{3}[(O_1+O_n)+4(O_2+O_4+…+O_{n-1})+2(O_3+O_5+…+O_{n-2})]\) |

**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 .

- Earthwork for highway, railway, retaining walls.
- Volume for reservoir capacity
- Concreting work
- Storage requirement

**Computation of Volume of Irregular Shape**

**1.Trapezoidal Rule / End Area Method**

This method is also called as end are method.

\(V=d[\frac{A_1+A_n}{2}+A_2+A_3+…A_{n-1}]\) |

**2.Prismoidal / Simpson’s Rule**

\(V=\frac{d}{3}[(A_1+A_n)+4(A_2+A_4+…+A_{n-1})+2(A_3+A_5+…+A_{n-2})]\) |

**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.

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