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Chapter: Civil Engineering - Surveying and Civil Engineering Materials

Calculation of Areas from offsets to a Base line

The area may be calculated by the rule: ü Mid-ordinate Rule ü Average ordinate Rule ü Trapezoidal rule ü Simpson’s Rule


Calculation of Areas from offsets to a Base line


The area may be calculated by the rule:

ü Mid-ordinate Rule

ü Average ordinate Rule

ü Trapezoidal rule

ü Simpson’s Rule

 

1.Mid-ordinate rule:

 

Area = (O1+O2+……….+on)d

 

O1, O2   = the ordinates at the mid points of each division

n           = number of divisions

L          = Length of base line= nd

d          = Distance of each division

 

2.Average ordinate Rule

 

Area  = ((O1+O2+……On)/(n+1))L

O1, O2….= Ordinates at the end of each division

 

3. Trapezoidal Rule

 

Area= (((O0+On)/2)+(O0+O2+……On-1))d

 

4. Simpson’s Rule

 

Area = ((d/3) x ((O0+On+4(O1+O3+On-1)+2(O2+O4+….+On-2))

 

 

Example: 1

 

The following perpendicular offsets were taken at 10 meter intervals from a survey line to an irregular boundary line.

 

3.15m, 4.3m, 8.2m, 5.6m, 6.85m, 7.6m, 4.2m, 5.6m, 4.3m

 

Calculate the area enclosed between the survey line, the irregular boundary line, and first and last offsets, by the application of

 

a)     Average ordinate rule

b)    Trapezoidal rule

c)     Simpson’s rule

Average     

d        = the interval between the offset = 10 m

n        = number of divisions    = 8

n+1 = number of ordinates      = 8+1 =9

L=  Length of the base line      = 8 x 10 = 80m

 

a)    Average ordinate rule

 

Area  = ((O1+O2+……On)/(n+1))L

O1, O2….= Ordinates at the end of each division

 

 

Area = (( 3.15 + 4.3 + 8.2+ 5.6 + 6.85 + 7.6 + 4.2 + 5.6 + 4.3 )/(8+1)) x 80 =442.66m2

 

b)    Trapezoidal rule

 

Area= (((O0+On)/2)+(O0+O2+……On-1))d

 

Area = (3.725 + 42.35) x10 =460.75m2

 

c)     Simpson’s rule

 

Area= 10/3(7.45 +92.4 +38.5)  = 461.167m2

 

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Civil Engineering - Surveying and Civil Engineering Materials : Calculation of Areas from offsets to a Base line |


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