Area & volume 3

Er. Mital Damani

INTRODUCTION

The term area in the context of surveying refers to the area of the tract of land projected upon the horizontal plane and not to the actual area of the land surface. Area may be expressed in the following units :

1 : Square meters 2 : Hectares (1 hectare =10000 m 2 =2471 acres)3 : Square feet4 : Acres (1 acre =4840 sq. yd. =43,560 sq. ft.)5 : Square kilometer (km2) = (1 km2 =106 m2)

Avg. oridnate rule

In this method, chain line is run approximately in the centre of the area to be calculated. With the help of the cross-staff or optical square,

Area of fig 1 can be calculated asA=A1+A2+A3+A4+A5A1=1/2(58-25)*10=165 m2 A2=1/2(25*10)=125 m2A3=1/2(16*12)=96 m2A4=1/2(12+9)*(50-16)=357 m2A5=1/2(58-50)*9=36 m2Total area of field A=779 m2

The area may be calculated in the following two ways.Case-1: considering the entire area:The entire area is divided into regions of a convenient shape and calculated as follows:(1) by dividing the area into triangles.(2) by dividing the area into squares.(3) by drawing parallel lines and converting them to rectangles.

Triangle area =1/2*base*altitudeArea=sum of areas of triangles

Each square represents unit area 1 cm2 or 1 m2Area=nos. of square *unit area

Area = Length of rectangle Constant depth

In this method, a large square or rectangle is formed within the area in the plan. The ordinates are drawn at regular intervals from the side of the square to the curved boundary.

Total area A=Middle Area A1+boundary area A2

Middle area can be subdivided into simple geometrical shapes , such as triangle rectangle , squares, trapezoids etc and Area of these figures are determined from the dimensions obtained from the plan.

1 The mid-ordinate rule2 The average ordinate rule 3 The trapezoidal rule 4 Simpson rule

The mid ordinate Rule :

Let O1,O2,O3,.,On=OrdinateAt equal intervals .

l=length of base line d=common distance between ordinatesh1,h2,.,hn=mid-ordinatesArea of plot=h1*d+h2*d++hn*d =d(h1+h2+.+hn)i.e. Area =common distance*sum of mid-ordinates

Lets O1,O2,..,On=Ordinate or offsets at regular intervals

L=Length of base line n= Number of divisions n+=number of ordinates

Area =O1+O2+.+On/n+1*l

=sum of ordinates/no. of ordinates *length of base line

While applying the trapezoidal rule, boundaries between the ends of the ordinates are assumed to be the straight. Thus, the area enclosed between the base line and the irregular boundary line are considered as trapezoids.

Let O1,O2,.On = Ordinates at equal intervals d=Common distance

1st Area =O1+O2/2*d2nd Area =O2+O3/2*d3rd Area = O3+O4/2*d last area = On-+On/2*d

Total area =[O1+2O2+2O3+2O4 +2On-1 +On]*d/2[O1+On+2(O2+O3+.+ On-1)]*d/2

Common distance/2 [(1st ordinate +last ordinate)+2 (sum of other ordinates)

In this rule the boundaries between the ends of ordinates are assumed to form an arc of the parabola. Hence Simpson rule is sometimes called the parabola rule.

Let O1,O2,O3=Three consecutive ordinates d=Common distance between the ordinates Area AF2DC = area of trapezium AFDC+ Area of * segment F2DEF Here,

Area of trapezium =O1+O2/2*2d

Area of segment =2/3*area of parallelogram F13D

= 2/3 *E2*2d =2/3 *{O2-O1+O2/2}*2d

So, the area between the first two divisions,

1=O1+O3/2*2d+2/3{O2-O1+O3/2}*2d=d/3(O1+4O2+O3)

Similarly, the area between next two divisions

2= d/3(O3+4O4+O5) and so on .

Total area =d/3 (O1+4O2+2O3+4O4++On)

=d/3 [o1+on+4(o2+o4+)+2(o3+o5+)]=common distance/3[(1st ordinate + last ordinate) + 4(sum of even ordinates) + 2 (sum of remaining odd ordinates)]

1. The boundary between the ordinates is considered to be straight 2. There is no limitation . It can be applied for any no. of ordinates.3. It gives an approximate result.

1.The boundary between the ordinates is considered to be an arc of a parabola .2. This rule can be applied when the no. of ordinates must be odd.3. It gives a more accurate result than the trapezoidal rule

A=M(FR IR +/- 10N + C)