# 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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### In a triangle, G is the centroid of triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB.

G is the centroid of the triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB. According to the solution manual, we can let the midpoint of AC be H. D, G, and H are collinear as G is ...
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### Coordinates of the centroid of volume

Determine the coordinates $$(x,y,z)$$ of the centroid of volume of this composite machine element. Here is the machine element: machine I know you have to use some integral with dV but I'm not sure ...
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### Centroid coordinates of an irregular quadrilateral within a rectangular plane

I am developing a robotic project; the robot moves within a rectangular area 80cm x 180cm on a level horizontal surface. The area is bounded by four vertical walls, the robot has onboard four ultra-...
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### How to determine that a cluster of points is actually two separate clusters?

I have a cluster of points defined by $x$ and $y$ coordinates. I know I can work out the center of the cluster by taking the average of $x$ and the average of $y$. Sometimes when I plot those points ...
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### In triangle $ABC$, $G$ is the centroid, $I$ is the incenter, $GI$ || $BC$, what is $\frac{AB+AC}{BC}$?

In triangle $ABC$, $G$ is the centroid, $I$ is the incenter, $GI$ || $BC$, what is $\frac{AB+AC}{BC}$? I have little to no idea what to do with this problem. I drew the diagram and called the point ...
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### Coordinates of centroid of a triangle

If $A(x_1,y_1) , B(x_2,y_2) , C(x_3,y_3)$ be the vertices of a triangle then prove that coordinates of centroid are given by $(\frac{(x_1 + x_2 + x_3)}3 , \frac{(y_1 + y_2 + y_3)}3)$
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### Centroid Math Notes

Was there a typo in the notes? I noticed that when calculating the center of mass for y they do not square twice as shown in the formula. Can anyone confirm this?
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### find centroid of hyperpyramid

I'm having trouble computing centroid of hyperpyramid (assume we have n points in n dimension). I already search a lot and I find how can calculate triangle and pyramid centroid, but I don't know how ...
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### Triangle inscribed in a circle,2 points fixed and 1 moving. The track of centroid makes a circle but how do I prove it without cartesian coordinate?

Triangle ABC and circle O. A and B are fixed, but C is moving on the circle. So I have triangle ABC and circle O. A and B are fixed on the circle, but C is moving around the circle. Let G is the ...
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### What is the centroid of the area bounded by x^3 and 10-x?

I understand what the problem is asking, but I don't fully understand the concept behind it. I know the formula for x bar is 1/A times the integral of x(f(x)-g(x)), and that the formula for y bar is ...
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### Constructing equilateral triangle from three equilateral triangles.

I have heard about this problem some time ago and don't remember all the details.Therefore problem's conditions may have some mistakes. Problem: We have three arbitrary (size of edges, location and ...
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### Finding the area of a triangle if the distances from the centroid to the vertices are $3$, $4$, $5$ [duplicate]

Let $G$ be the centroid of $\triangle ABC$, and let $AG=3$, $BG=4$, $CG=5$. Find the area of $\triangle ABC$. So I calculated the lengths of the diagonals which is $9$, $12$, $15$, but now I don't ...
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### Get vertex value that shapes 90 degrees with triangle centroid

I have this triangle: I'm trying to get the vertex value highlighted in the green circle in order to draw that red line. Is there any equation that I can use to extract that value? ...
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### Centroid of area enclosed by $x^n$ and $x^{1/n}$.

I am interested in finding the centroid of the surface enclosed by the lines $y=x^n$ and $y=x^{1/n}$ where $n \in \mathbb N, n >1$. The exercise, gives the following sketch. I know that in order ...
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### Why is k proportional to centroid linkage distance mean and variance in k-means?

I've noticed that if I'm doing k-means clustering (in MATLAB) on any set of data, the mean and variance in centroid linkage distance appears to always be proportional to k. Is k always proportional ...
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### Is the number of convex regions proportional to the inter-region centroid variance?

Say I completely divide some euclidean space into n convex regions for different values of n. Veronoi cells would be a good example. Say I take a subset of m regions. It seems almost obvious to me ...
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If $A(\cos\alpha, \sin\alpha)$, $B(\sin\alpha, -\cos\alpha)$ and $C(1,2)$ are the vertices of $\triangle ABC$, find the locus of the triangle's centroid as $\alpha$ varies. Let centroid be $(h,k)$, $$... 1answer 29 views ### Centroid of a plane figure of a plane figure. A plane figure is enclosed by the parabola y^2 =4x and the line y=2x. Determine the position of the centroid of the figure. Here is what I tried : Plotted the graph. y=4ax, y=2x \\ 4x(x-1)=0 \... 0answers 100 views ### Finding centroid's coordinates using Pappus theorem The task is to find the centroid of the given triangle (see the image above). We also should use the fact that the volume of a cone of radius r and height h is V = \frac{1}{3}\Pi r^2h. My ... 2answers 197 views ### How to use Vectors to find the centroid of a tetrahedron? [closed] Suppose that four points - A,B,C,D (with position vectors a,b,c,d) are the vertices of a tetrahedron. And the mid points of BC, CA, AB, AD, BD, CD are denoted by P,Q,R,U,V,W. Using these info, I ... 2answers 150 views ### The centroid of a set lies in its convex hull I've come across the following claim on a Wikipedia page about the centroid: The geometric centroid of a convex object always lies in the object How should I prove this claim for convex subsets of ... 1answer 26 views ### Area moments of a Freeman chain contour The area moments of a polygon can be computed by generalizations of the shoelace formula for area. In particular, the first and second order moments are given by https://en.wikipedia.org/wiki/... 1answer 135 views ### Finding Centroid of a curve$$x=e^t , y = \sqrt{2} t , z= \ e^{-t} ; ~~ ~~ 0\leq t \leq 1$$Find the centroid of this curve. I know one formula for centroid which is x = \frac{\int {x dx dy}}{\int dx dy} and for y as well, ... 4answers 45 views ### Two parallelograms and a centroid A parallelogram ABCD is given. The points M_1 and M_2 are on AC and M_3 and M_4 are on BD such that AM_1 = CM_2 = \frac{1}{3}AC and BM_3=\frac{1}{3}BD. If M_1M_3M_2M_4 is a ... 2answers 59 views ### An identity associated with the centroid of a triangle Four year ago, I am looking for a proof of my identity as follows: Let ABC be a triangle, let G be the centroid of ABC. Let P be any point on the plane. Let H, N, O on the plane such that: ... 1answer 123 views ### How do I find the moments and center of mass of a Lamina with given density? I am trying to solve for the center of mass of the shape given below. I started by finding the area of the shape, which is A = Atriangle + Acircle/4  = 1/2 + pi/4 =(2+pi)/4 then I used the ... 4answers 104 views ### Calculate coordinates of a centroid of ABC We have a triangle ABC, where on the cartesian coordinate system: A lies on [-3, -2], B lies on [1, 1], C lies on [0, -6]. How do we calculate coordinates for the centroid of this ... 2answers 38 views ### Proof that the triangle formed by mass centers is equilateral. Let ABC be a triangle and consider A_1, B_1, C_1 outside the triangle such that triangles ABC_1, BCA_1 and ACB_1 are equilaterals. Consider now A_2, B_2 and C_2 mass centres of ... 3answers 149 views ### Area of the intersection of two triangles. Let \triangle{ABC} be a triangle with AB=5, BC=7, and CA=4. Define D, E, and F, to be the midpoints of AB, BC, and CA respectively. Let G the intersection of the medians of \... 1answer 95 views ### Intuitive Explanation to Pappus Theorem Pappus's theorem is as follows: First theorem: The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on ... 1answer 428 views ### 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 ... 1answer 51 views ### centroid algorithm robust to missing poins I need to find a center point of a person given the coordinates of all the joints. The joints of a person can be represented as a nodes of a graph with a fixed structure. The catch is some of the ... 2answers 79 views ### z_1^2+z_2^2+z_3^2=3z_0^2 if z_1,z_2,z_3 be the vertices of an equilateral triangle and z_0 be the circumcentre Prove that, if z_1,z_2,z_3 be the vertices of an equilateral triangle and z_0 be the circumcentre, then z_1^2+z_2^2+z_3^2=3z_0^2 My Attempt$$ z_0=\frac{z_1+z_2+z_3}{3}\implies 3z_0=z_1+z_2+z_3\...
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Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there ...
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### How to find the best fit point inside a cluster?

I have a cluster with many points. Like this: Where I can visually identify a cluster of points and a ...
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### Volumetric center of a polygon?

Is there a quick and efficient algorithm for calculating the volumetric center (probably the wrong term) of a polygon like that shown in the figure below somewhere around the blue dot? I'm not ...
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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 $[a, b]$, $-\infty<a<b<\infty$. Best known ...