Editing Centroid (section)

=== By geometric decomposition ===
The centroid of a plane figure <math>X</math> can be computed by dividing it into a finite number of simpler figures <math>X_1, X_2, \dots, X_n,</math> computing the centroid <math>C_i</math> and area <math>A_i</math> of each part, and then computing

<math display=block>
C_x = \frac{\sum_i {C_i}_x A_i}{\sum_i A_i}, \quad
C_y = \frac{\sum_i {C_i}_y A_i}{\sum_i A_i}.
</math>

Holes in the figure <math>X,</math> overlaps between the parts, or parts that extend outside the figure can all be handled using negative areas <math>A_i.</math> Namely, the measures <math>A_i</math> should be taken with positive and negative signs in such a way that the sum of the signs of <math>A_i</math> for all parts that enclose a given point <math>p</math> is <math>1</math> if <math>p</math> belongs to <math>X,</math> and <math>0</math> otherwise.

For example, the figure below (a) is easily divided into a square and a triangle, both with positive area; and a circular hole, with negative area (b).
{{multiple images
|align=center
|height=200
|direction=horizontal
|image1=COG 1.png
|caption1=(a) 2D Object
|image2=COG 2.png
|caption2=(b) Object described using simpler elements
|image3=COG 3.png
|caption3=(c) Centroids of elements of the object
}}
The centroid of each part can be found in any [[list of centroids|list of centroids of simple shapes]] (c). Then the centroid of the figure is the weighted average of the three points.  The horizontal position of the centroid, from the left edge of the figure is
<math display=block>x = \frac{5 \times 10^2 + 13.33 \times \frac{1}{2}10^2 - 3 \times \pi2.5^2}{10^2 + \frac{1}{2}10^2 -\pi2.5^2} \approx 8.5 \text{ units}.</math>
The vertical position of the centroid is found in the same way.

The same formula holds for any three-dimensional objects, except that each <math>A_i</math> should be the volume of <math>X_i,</math> rather than its area.  It also holds for any subset of <math>\R^d,</math> for any dimension <math>d,</math> with the areas replaced by the <math>d</math>-dimensional [[measure (mathematics)|measure]]s of the parts.