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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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