# Questions tagged [centroid]

"The centroid or geometric center of a plane or solid figure is the arithmetic mean ("average") position of all the points in the shape. " This tag is for questions about the centroid of a geometrical shape, its properties and computation.

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### How to find distance between center of rectangle and bound with degree?

Given width and height of rectangle, coordinates of centroid and degree, how to find coordinates of a point extended from centroid with degree? In the below image, I want to calculate (??, ??). Any ...
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### How to calculate the centroid of a Polytope?

Given a polytope is divided into simplexes, is it correct to calculate the centroid of the polytope as the average sum of its simplex centroid coordinates
1 vote
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### n-fold centered tensor analog to double centered matrix?

A double centered matrix has the properties, that: the mean over all components the mean of each column the mean of each row is 0 (in every component). To double-center any given matrix $M$ with ...
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1 vote
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### Prove that the area centroid of a planar region in $\mathbb{R}^3$ projects onto that of its projection onto the $xy$-plane

Consider an arbitrary convex bounded region $\Omega'\in\mathbb{R}^2$ which is the projection onto the $xy$-plane of a region $\Omega$ that lies on a plane in $\mathbb{R}^3$ (that is not parallel to ...
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### Centroid of connected 3D shapes

This might be a dumb question, but I tried to search for answers online and couldn't find any. I couldn't find too much information about centre of mass of irregular 3D objects in general. So, I have ...
1 vote
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### Explanation for the formula for the centroid $\bar{x}$

First, can someone provide a simple explanation for the $\bar{x}$ formula: $$\frac{\iint xdA}{area}$$ My understanding of the formula is as follows: we let $z=x$, calculate the volume, and divide by ...
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### Under what conditions does every line through the centroid divide in half?

Let $G$ be an enclosed region of the complex plane, such that the property $\iint_G z dA=0$ holds; in other words, such that the center of mass is the origin. Draw an arbitrarily line $C$ passing ...
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### The point of intersection of the four space diagonals of a general parallelepiped, the centroid

I know that the centroid of a parallelogram is the intersection of the diagonals, but is it true that the centroid of a parallelepiped is the intersection of the space diagonals? I'm talking about the ...
1 vote
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### Volume of a rotated ellipse by x=y

I'm calculating a volume of a rotated ellipse by the line $x=y$ using Pappus Theorem, the ellipse has an equation of : $$(9x^2/16)+(36y^2/25)=1$$ Using Pappus Theorem, I can just plug in the length of ...
1 vote
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### Centroid of the region bounded by y = f(x) and x = f(y) curve [closed]

How do i find the centroid of the region if $x = f(y)$ is involved ? For example : find the centroid of the region bounded by $y = x^2$ and $x = y^2$ And how do i find it if there is curve ...
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### Petr-Douglas-Neumann theorem - centroids don't match

According to Petr-Douglas-Neumann theorem (Wikipedia article) after all n-2 steps (n-number of sides/vertices in the polygon) are complete, the result is a regular polygon whose centroid coincides ...
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### How would one define the "center" of a closed, non-discrete 1d loop?

Centroids describe the average of all the discrete points on a curve (in 2d space), but why is it that the center of a circle or ellipse is what it is? Moreover, is it possible to determine the center ...
1 vote
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### Find the center of mass of a quarter circle given by $\sqrt{r^2−x^2}$, $x\in[0,r]$
Q: The center of mass of the quarter circle given by $y=\sqrt{r^2−x^2}$, $x\in[0,r]$ is the point $(x, y)$. My first thought was that the circle was bounded by the points $(0,0)$, $(0,r)$, $(r,0)$ and ...
Let $O$ be the circumcenter of the regular polygon $P_n$. Then for any $A\in P_n$ one has $d(A,O)\leq r$, where $d$ is the usual Euclidean distance and $r$ is the polygon's circumradius. Prove that ...