Aquaculture Engineering (2+1)

Calculation of area of irregular plane surfaces - Trapezoidal rule

Unit 6- Calculation of area and volume

Calculation of area of irregular plane surfaces

Calculation of area of irregular surfaces can be calculated using the following formulae.

The trapezoidal rule
A trapezoid is a plane four sided figure, having two sides parallel to each other. If the lengths of these sides are Y₁ and Y₂, and they are “h” apart, the area of the trapezoid is given by,

A = h/2 (Y₁+Y₂)

Let us consider a curvilinear figure. It can be divided into number of approximate trapezoids by covering it with “n” equally spaced ordinates, which are at a distance “h” apart. The breadth of the ordinates are in order being Y₁, Y₂, Y₃, Y₄…………………Y_n .

Commencing from the left hand side the area of each trapezoid are given by
a₁= ½ h (Y₁+Y₂)
a₂ = ½ h (Y₂+Y₃)
a₃ = ½ h (Y₃+Y₄)
a₄ = ½ h (Y₄+Y₅)
a₅ = ½ h (Y₅+Y₆)
a₆ = ½ h (Y₆+Y₇) and so on
By addition the total area A of the figure is given by
A= a₁ +a₂ + a₃ +a₄ + a₅ + a₆ +...............
A = ½ h (Y₁+Y₂) + ½ h (Y₂+Y₃) + ½ h (Y₃+Y₄) + ½ h (Y₄+Y₅) + ½ h (Y₅+Y₆) +½ h (Y₆+Y₇)
A = ½ h (Y₁+2Y₂+2Y₃+2Y₄+2Y₅+2Y₆+……………..+Y_n)
A = ½ h [Y₁+Y_n +2 (Y₂+Y₃+Y₄+Y₅+……………..+Y _n-1)]

This is termed as TRAPEZOIDAL RULE, the more numerous the ordinates; the more accurate will be the answer.