Calculation of area of irregular surface using Simpsons method

Methods of Calculation
of Areas in Surveying |

Simpson’s Rule

Calculation of Areas in Surveying | Simpson’s Rule
In one of my previous articles, I discussed Midpoint
Ordinate Rule and Average Ordinate Rule in detail with an
example and listed out various important methods used
for the calculation of areas in Surveying. In this article,
we will deal with the next important method (rule) i.e.
Simpson’s Rule along with a numerical example used for
the calculation of areas in the field of Surveying.
Here are the five important rules (Methods) used for the
calculation of areas in Surveying:

1. Midpoint ordinate rule
2. Average ordinate rule
3. Simpson’s rule
4. Trapezoidal rule
5. Graphical rule

Simpson’s Rule
Statement

It states that, sum of first and last ordinates has to be
done. Add twice the sum of remaining odd ordinates and
four times the sum of remaining even ordinates. Multiply to
this total sum by 1/3rd of the common distance between the
ordinates which gives the required area.
Where O 1, O 2, O 3 , …. O n are the lengths of the
ordinates
d = common distance
n = number of divisions
Note:

This rule is applicable only if ordinates are odd, i.e. even
number of divisions.
If the number of ordinates are even, the area of last
division maybe calculated separated and added to the
result obtained by applying Simpson’s rule to two
remaining ordinates.
Even if first or last ordinate happens to be zero, they are
not to be omitted from Simpson’s rule.
The following offsets are taken from a chain line to an
irregular boundary towards right side of the chain line.

Chainage 0       25      50.         75     100    125   150

Offset ‘m’ 3.6   5.0      6.5        5.5   7.3.     6.0      4.0

Common distance, d = 25m

Area = d/3[(O 1+O 7) + 2 (O 3 +O 5)+4(O 2+O4+O 6)]
= 25/3[(3.6+4)+2(6.5+7.3)+4(5+5.5+6)]
Area = 843.33sqm