Areas
 Trapezoidal Rule
 Simpson's Rule
 Other Methods
 Coordinates
 Example

Coordinates

It is required to determine the area of the closed polygon 1,2,3,4,5,1. For each of the corners the X and Y coordinates are known. A solution may be found by constructing a set of trapezoids bounded by the X axis, the individual sides of the polygon and two sides parallel to the Y axis. These trapezoids can be seen in the figure below and are labeled 12ca1, 23ec2, 34de3, 45bd4 and 51ab5. The area of the polygon is the sum of the areas of the individual trapezoids.

Working around the polygon in an anti-clockwise direction the sum of the individual areas is;

2AT =

[Y1X2 + Y2X2 - Y1X1 - Y2X1      +     Y2X3 + Y3X3 - Y2X2 - Y3X2      +     Y3X4 + Y4X4 - Y3X3 - Y4X3      +     Y4X5 + Y5X5 - Y4X4 - Y5X4      +     Y5X1 + Y1X1 - Y5X5 - Y1X5]

Collecting like terms and cancelling out the underlined terms leads to the simplified equation below,

2A_T = [Y₁X₂ - Y₂X₁] + [Y₂X₃ - Y₃X₂] + [Y₃X₄ - Y₄X₃] + [Y₄X₅ - Y₅X₄] + [Y₅X₁ - Y₁X₅]

If the terms are collected as above the formula may be represented in the general form of,

2A = S[X_i+1Y_i - X_iY_i+1]

The resultant area will be positive or negative depending on whether a polygon is traversed in a clockwise or anti-clockwise direction. The amount of computations maybe significantly reduced if the coordinates of the first point are reduced to zero; that is X₁ = Y₁ = 0.
This may be achieved by the simple transformations,

x_i= X_i - X₁ and y_i = Y_i - Y₁

For a four sided polygon this will reduce the number of multiplications from eight to four.

Calculating The Required Precision For Measuring An Irregular Area

Before taking measurements to calculate the area of an irregular shape, the formula below is useful to check the expected precision of a computed area. It can be shown from propagation of error that the precision of an area can be approximated by the formula :

area precision = 0.5 . n . average side length . coordinate precision

where: n = the number of sides of the area