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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.

The centroid or geometric center of a plane or solid figure is the arithmetic mean ("average") position of all the points in the shape. The notion generalizes immediately to n-dimensional figures.

If a solid body has uniform mass density, the centroid agrees with its center of mass.

This tag is for questions about the centroid of a geometrical shape, properties and computation.

Use the tag if the question relates to the geometrical properties of the centroid.

Use the or tag if the question relates to finding the centroid of an object by integration.

299 questions
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Sphere equation given 4 points

Find the equation of the Sphere give the 4 points (3,2,1), (1,-2,-3), (2,1,3) and (-1,1,2). The *failed* solution I tried is kinda straigh forward: We need to find the center of the sphere. Having the points: …
danielrs
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Point that divides a quadrilateral into four quadrilaterals of equal area

Consider an irregular quadrilateral $ABCD$. Let $E,F,G,H$ be the midpoints of its edges. It seems that there is a point $K$ such that $$ S_{AHKE} = S_{EKFB} = S_{KHDG} = S_{KGCF} \left(= \frac{1}{4} S_{ABCD}\right) $$ I'm curious whether the point…
uranix
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can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex at maximum distance from c. The following picture…
Don Hatch
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Centroid within non-convex 2d polygon

The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself: Is there a definition of a centroid-like point which always lies within the…
pschill
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Can a spiral have its centroid at the origin?

A spiral is a curve $\gamma$ with the polar equation $r=f(\theta)$ where $f$ is a continuous positive strictly monotone function on some interval $[a, b]$, $-\infty
user357151
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Prove that $M$ is the centroid of the triangle $BCD$.

the question Consider the tetrahedron $ABCD$ and $M$ a point inside the triangle $BCD$. Parallels taken from $M$ to the edges $AB$, $AC$, $AD$ intersect the faces $(ACD)$, $(ABD)$, respectively, $(ABC)$ at the points $A', B',$ respectively, $ C'$.…
IONELA BUCIU
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Mathematical Paradox: How Can The Center of a Shape Be Located OUTSIDE This Shape?

Recently I have been learning about Geospatial Analysis in which we are often interested in using computer software to analyze the mathematical properties and characteristics of polygons (e.g. calculating their centroids). For example, using the R…
stats_noob
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The centroid of the zeros of the kth partial sum of exp(z) is -1?

The question is the next: Let $P_k=1+z+\frac{z^{2}}{2!}+...+\frac{z^{k}}{k!}$, the kth partial sum of $e^{z}$. (a) Show that, for all values of $k\geq 1$, the centroid of the zeros of $P_{k}$ is -1. (b) Let $z_{k}$ be a zero of $P_{k}$ with maximal…
Magic Unicorn
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Articles on the "Property I found" and other types of Centers (excluding the Centroid)?

I am a first-year undergraduate student. I came up with a kind of center property which I cannot find in articles online. I found the center on my own but needed help of mathematicians (@Rahul) on this site. Currently I'm unsure whether it applies…
Arbuja
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What is the Centroid of $z=\frac{1}{(1-i\tau)^{i+1}},\ \ \tau\in (-\infty,\infty)$

Some years ago I developed the closed curve (Ref. 1) $$z=\frac{1}{(1-i\tau)^{i+1}}, \ \ \tau\in (-\infty,\infty)$$ I was able to calculate the arc length and area, $$ s=\int |\dot z| d\tau=2\sqrt{2}\sinh\left(\frac{\pi}{2}\right)\\ A=\frac{1}{2}\int…
Cye Waldman
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Centroids of a polygon

Obviously, for any polygon we can define at least $3$ different centroids: $C1:\;$ mass center of the lamina; $C2:\;$ mass center of vertices with equal masses; $C3:\;$ mass center of the perimeter. For the triangle $C1 = C2 \ne C3$; for common,…
lesobrod
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^2 dz=\pi \int_0^h \frac{R^2}{h^2}z^2 dz=\pi…
Emilio Novati
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Show for the centroid $M$ of a triangle $ABC$ that $\vec{OM}=\frac13(\vec{OA}+\vec{OB}+\vec{OC})$

Show for the centroid $M$ of a triangle $ABC$ that $$\vec{OM}=\dfrac13\left(\vec{OA}+\vec{OB}+\vec{OC}\right)$$ where $O$ is an arbitrary point. I haven't studied position vectors and am very new to vectors so I would like a simple solution. Any…
Math Student
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Finding centroid of spherical triangle

Given three vectors $u,v,w\in S^2$, and a spherical triangle $[u,v,w]$, find its centroid, i.e. the point of intersection of the three medians $\left[u, \frac{v+w}{|v+w|}\right]$, $\left[v, \frac{u+w}{|u+w|}\right]$ and $\left[w,…
sequence
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Proof-validation of centroid's existence

So, a friend of mine came up with this unorthodox proof of the centroid's existence so I figured I could share it here so that someone can confirm that it's a fine one. I think it is correct, but I haven't seen anything like it. I will write it…
Martín Forsberg Conde
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