# Centroid of Two Rectangles

- Author:
- Dr. Doug Davis, 3D

- Topic:
- Centroid or Barycenter, Rectangle

## Centroid

First Move Point D to (0,1)
The centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. (https://en.wikipedia.org/wiki/Centroid)
If the area was a thin plate parallel to the earth surface, the centroid would be at the center of gravity. The center of gravity is the point where a shape would balance. The formula for the centroid can be expressed as a ratio of integrals, and where is the centroid point and the integrals are over the area divided into differential area elements . Tables of centroids of common shapes can be used. Because of the additive property of integrals, the centroid of the combination of several basic shapes can be calculated as and where is the centroid of each basic shape and is the area of each corresponding shape. Shapes can also be subtracted by using a negative area.
The best way to do these calculations is with a table or spreadsheet. Here a spreadsheet is used to calculate the centroid of two rectangles. The centroid of a rectangle is in its center. The sum (total) of the table columns of , and are respectively the terms , , and in the formula above for and .

## Description

This applet computes the centroid of two rectangles using a spreadsheet as described above. The two rectangles are defined by the points A and B for Area 1 and C and D for area 2. If area 2 overlaps area 1 it is converted into a negative area of the intersection of the rectangles. This way more interesting combined shapes can be formed. The centroids of each area are shown as pluses, + and the combined centroid is shown as a cross, x.

## Activities

Move the points around and note how the centroids change. Note that the combined centroid is closer to the larger areas centroid.
Compare the centroids made by combining the rectangles with points A=(-3,1), B=(-1,4),C=(-1,1) and D=(0,2) with the centroids made by subtracting rectangles with points A=(-3,1),B=(0,4), C=(-1,2) and D=(0,4).