General formula for centre of mass of polygons

In summary, the centre of mass of a polygon can be found using the formula (x̄, ȳ), where x̄ is the average of the x-coordinates and ȳ is the average of the y-coordinates of the polygon's vertices. For a regular polygon, the centre of mass is located at the intersection of the medians. It can be located outside of the polygon in irregular or non-uniform cases. The centre of mass is significant in physics and engineering, and has various real-world applications such as determining balance and stability, analyzing motion, and designing efficient vehicles.
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The position vector of the center of mass of a triangle is ##\frac{1}{3}(\mathbf{a}+\mathbf{b}+\mathbf{c})##.

Screen Shot 2016-07-16 at 4.08.19 am.png


Is the position vector of the center of mass of a planar four-sided figure ABCD ##\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})##? Does this generalise to n-sided figure?

How about a 3D figure ABCD in the form of an arbitrary triangular pyramid (whose faces need not be equilateral)? Is the center of mass also ##\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})##? Does this generalise to n-dimensonal figure?
 
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1. What is the general formula for finding the centre of mass of a polygon?

The general formula for finding the centre of mass of a polygon is (x̄, ȳ), where x̄ is the average of all the x-coordinates of the polygon's vertices and ȳ is the average of all the y-coordinates of the polygon's vertices.

2. How is the centre of mass of a regular polygon calculated?

For a regular polygon with n sides, the centre of mass is located at the intersection of the n medians, where each median connects a vertex to the midpoint of the opposite side.

3. Can the centre of mass of a polygon be located outside of the polygon?

Yes, the centre of mass of a polygon can be located outside of the polygon if the polygon is irregular or has a non-uniform distribution of mass.

4. What is the significance of the centre of mass of a polygon?

The centre of mass is an important concept in physics and engineering as it describes the average position of mass in a system. It is also used to calculate the overall motion and stability of objects.

5. How can the centre of mass of a polygon be used in real-world applications?

The centre of mass of a polygon can be used in various real-world applications such as determining the balance and stability of structures, analyzing the motion of objects, and designing efficient aircraft and vehicles.

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