What is Centroid: Definition and 178 Discussions

In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions.While in geometry the word barycenter is a synonym for centroid, in astrophysics and astronomy, the barycenter is the center of mass of two or more bodies that orbit each other. In physics, the center of mass is the arithmetic mean of all points weighted by the local density or specific weight. If a physical object has uniform density, its center of mass is the same as the centroid of its shape.
In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center.

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  1. Nova_Chr0n0

    Engineering Centroid of Composite Bodies - Statics of Rigid Bodies

    The figure and formulas is shown above. My strategy of cutting the areas/shapes is shown below: Area 1 = Area of Triangle Area 2 = Area of the square - Area of the quarter circle Area 3 = Area of the larger quarter circle - Area of the smaller quarter circle Computing for the areas, I got...
  2. A

    Centroid calculation using integrals

    Hello! Im given this function ## f:[-\pi/2,1] -> [0,1]## with f(x) = 1-x for x (0,1] and f(x) = cos(x) for x ##[-\pi/2,0] ## And im susposed to find the centroid of this function so xs and ys. For that I am given these 2 equations ( I found them in the notes) ## x_s =\frac{1}{A}...
  3. A

    I Finding the center of area (centroid) of a right triangle

    To find the y value of the centroid of a right triangle we do $$\frac{\int_{0}^{h} ydA}{\int dA} = \frac{\int_{0}^{h} yxdy}{\int dA}$$ What is wrong with using $$\int_{0}^{h} ydA = \int_{0}^{b} y*ydx$$ as the numerator value instead especially since ydx and xdy are equal and where h is height of...
  4. Tapias5000

    How can I locate the coordinates of the centroid of a cone in Z?

    This is the picture of the problem. My solution is: I'm not sure if the limit is 0 to 2 or 0 to 4...
  5. kshitij

    How to find centroid of a hemisphere using Pappus's centroid theorem?

    I recently learned how to calculate the centroid of a semi-circular ring of radius ##r## using Pappus's centroid theorem as ##\begin{align} &4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\ &d=\frac {2r}{\pi}\nonumber \end{align}## Where ##d## is the distance of center of mass of the ring from its base...
  6. S

    MHB Proving isosceles using centroid and medians

    I can definitely do this in the opposite direction, but
  7. S

    MHB Prove Centroid Goes Through It

    I can see how it would go through the centroid, but I don't know how to prove that it HAS to go through the centroid.
  8. J

    Engineering Finding Centroid of 4 Sections: Help Needed

    I found the centroid of four sections, 2 semicircles, 1 rectangle (xz plane) and the remaining right triangle. I tabulated the individual centroids (x,y,z) of the 4 sections, multiplied each one with the Area of respective section. In the end I calculated the required (x,y,z) by dividing (sum of...
  9. J

    Trivial : Use of "negative sign" when calculating centroid

    I realize that this is to be solved by breaking up the object into simple objects and using their known center of mass to find the center of mass of the entire object. 1. In the solution the circular gap is also considered in the calculations with a negative center of mass, why is this done? 2...
  10. P

    Polar coordinates of the centroid of a uniform sector

    If I use cartesian co-ordinates, I get: ##\bar{x}=\frac{1}{A}\iint x\, dA=\frac{1}{A} \iint r^2\cos\theta\, dr\, d\theta= \frac{2a\sin\theta}{3\theta}## ##\bar{y}=\frac{1}{A}\iint y\, dA=\frac{1}{A}\iint r^2\sin\theta\, dr\, d\theta= \frac{2a(1-\cos\theta)}{3\theta}## But if I use polar...
  11. E

    Finding the Centroid of an arc, and then of a sector, with heuristic arguments

    The first part is not a problem, I let one radius lie along the ##x## axis and then we can write down ##S_x = \frac{M}{2}f(\frac{\alpha}{2})\cos{\frac{3\alpha}{2}} + \frac{M}{2}f(\frac{\alpha}{2})\cos{\frac{\alpha}{2}} = Mf(\alpha)\cos{\alpha}## from which we can then get the following after...
  12. G

    Determining a centroid

    Summary:: I'm solving an exercise. I have the following center of gravity problem: Having the function Y(x)=96,4*x(100-x) cm, where X is the horizontal axis and Y is the vertical axis, ranged between the interval (0, 93,7) cm. Determine: a) Area bounded by this function, axis X and the line...
  13. T

    Defining the Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes

    Homework Statement: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes Homework Equations: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes Hello all; I am trying to understand the terms:- - Centroid for a 2D shape and 3D shape - Centre of Mass for a 2D...
  14. PainterGuy

    I Centroid of homogeneous lamina region R and the factor of "1/2"

    Hi, In one of the standard calculus textbooks, source #1, the formula for y-coordinate of center of gravity for a homogeneous lamina is given as follows. In another book of formulas, source #2, the formula is given without the factor "1/2" as is shown below. Personally, I believe that source...
  15. Zack K

    Centroid of an isoceles triangle

    Homework Statement Where is the center of mass of an isoceles triangle? Homework Equations xcm=∫xdV/V (where V is the volume of the triangle) The Attempt at a Solution The representation of the sides is what I'm confused with. Flipping the triangle to it's side is what's recommended to be...
  16. JessiLim

    Static and Kinetic Friction vs Normal Reaction Graphing

    How do i suppose to determine the uncertainty for the slope of my Static friction against normal reaction graph? My data for static friction and normal force has the uncertainty of +/- 0.0001 The uncertainty is too small for me to draw airbox/bar in the graph to draw the max and min slope...
  17. Sune Irl

    Calculus 2 - Center of Mass and Pappus Centroid Theorem

    Homework Statement determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices (1,0), (0,0), (1,1). Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x = -2. Homework...
  18. CivilSigma

    Determining the Centroid of a 3D Section

    Homework Statement [/B] I am having a problem understanding a calculation performed as part of a bigger solution in the design of Slabs. That is, how to determine the centroid of the critical shear section, which consist of 3 planes intersecting to form a 3D model (please see attached...
  19. karush

    MHB 15.6.19 Find the mass and centroid

    Find the mass and centroid of the following thin plate assuming constant density Sketch the region corresponding to the plate and indicate the location of the center is the mass The region bounded by $$y=ln x$$ $$x-axis$$ $$x=e$$ \begin{align}\displaystyle...
  20. T

    Geometry: Finding a Side Length in Triangle Using Centroid

    The Problem is #16 in the attached picture. Essentially, I need to find the length of BC using information about congruency and the location of the centroid. I've been able to show a whole bunch of things, but nothing that gets me close to actually finding out the missing side length. I began...
  21. M

    MHB Coordinates of the centroid

    Hey! :o We have a triangle $ABC$ with $A(a_1, a_2)$, $B(b_1, b_2)$ and $C(c_1,c_2)$. I want to show that the coordinates of the centroid S is $\left (\frac{1}{3}(a_1+b_1+c_1),\frac{1}{3}(a_2+b_2+c_2) \right )$. $S$ is the intersection point of the midpoints of AB, BC and CA. We have that...
  22. F

    Is my Centroid Calculation Correct?

    Homework Statement For this shape , it's clear that the centroid and the horizontal line of equal axis lies on the same horizontal line , am i right ? Homework EquationsThe Attempt at a Solution I'm not sure . correct me if i am wrong . [/B]
  23. S

    How to Determine Centroid of Gate Hydrostatics

    Homework Statement In this image : http://ezto.mheducation.com/13252703414204806874.tp4?REQUEST=SHOWmedia&media=2.83qs.jpg Why does the weight of the gate have a centroid at 2R/pi away from the force F? Homework Equations The centroid for a half circle in both x and y directions is =...
  24. D

    Area between 2 curves, Volume around X and Y, Centroid

    g(x)= √(19x) = upper curve f(x)= 0.2x^2 = lower curve Firstly, I found the point of intersection, which would later give the upper values for x and y. x=7.802 y=12.174 Then I found the area under g(x) and took away the area under f(x) to get the area between the curves. 31.67 units^2 This is...
  25. M

    Coordinate geometry - centroid (SL LONEY exercise problem)

    Homework Statement If G be the centroid of ΔABC and O be any other point, prove that , ## 3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2## ##and,## ##OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2## Homework Equations i m practising from S L LONEY coordinate geometry first chapter ... only the equation...
  26. F

    Twisting at shear center / centroid

    In the notes , the author stated that when the force is applied through the centorid of cross section , the channel will bend and twist. but , on the second page , the author stated that the shear center lies on an axis of symmetry of member's cross sectional area... So, i am confused whether...
  27. F

    Twisting at shear center / centroid

    Homework Statement In the notes , the author stated that when the force is applied through the centorid of cross section , the channel will bend and twist. but , on the second page , the author stated that the shear center lies on an axis of symmetry of member's cross sectional area... So, i am...
  28. MyNameIsNicholas

    How do you "read" this formula?

    Homework Statement No actual work, could just use some assistance in understanding formulas involving the centroid of an object, specifically with integrals. For example, how would you understand the following formula(s) (as seen in part 2)? I understand that the centroid is the sum of all the...
  29. F

    Shear stress at centroid vs other point

    [ Mod Note: moving this to Physics H/W ] Homework Statement for this question , I'm having problem with the shear stress at point E and shear stress at centorid. normally , the shear stress at the centoid will be maximum . But , in my working , I found that the shear stress at the centroid is...
  30. C

    Centroid of solid enclosed by surface z= y^2 , plane x=0 ,

    Homework Statement Find the centroid of solid enclosed by surface z= y^2 , plane x=0 , x = 1 and z =1 . The density is 1 Homework EquationsThe Attempt at a Solution Here's my working . Centoird = mass of inertia / mass So , i find the mass first . It's clear that the circle is on zx...
  31. H

    I Prove Pappus's centroid theorems without calculus

    Pappus's centroid theorems were discovered 17 centuries ago, when calculus wasn't invented yet. How are these theorems proved without using calculus? "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...
  32. Manwe Sulimo

    How do you find the centroid of this?

    Homework Statement Find the centroid of the shape formed by the equation y2=x3-x4, the x-axis, and the y-axis. Homework Equations A=∫f(x)dx Mx=∫(1/2)[f(x)]2dx My=∫x[f(x)]dx The Attempt at a Solution I'm stuck on the integral. I attempted u-substitution and got du=(1/2)(x3-x4)-1/2dx; "parts,"...
  33. CivilSigma

    What does it mean to find the centroid of a shape?

    Hello, I am currently studying how to find the centroid of shapes. And I understand that to find the location of the centroid, we must analyze the distribution of the mass over the x and y-axis (i.e calculating Qx and Qy). However what baffles me is that, given an L shaped beam, the centroid...
  34. R

    Centroid Question: Understanding & Measuring Distance

    I'm having trouble understanding what this question is actually asking for. Is it assuming the centroid to be the origin and asking how far the bottom of the shape extends downwards for the origin to be the centroid?
  35. W

    Axes passing thru centroid

    Homework Statement determine the second moment of inertia about the horizontal axis and vertical axis for the shaded area with respect to x and y axes through the centroid of the area . Homework EquationsThe Attempt at a Solution Since the x and y axes is drawn thru centroid , why not the y...
  36. W

    Derivation of formula of centroid

    Homework Statement what is y ′and y bar ? why y ′ is changed to y bar ? why are they = 0 ? Homework EquationsThe Attempt at a Solution
  37. W

    Centroid of composite area

    Homework Statement As in the photos Homework EquationsThe Attempt at a Solution my working is (6.43x10^-3)(310+10) + (11x10^-3)(155) / (6.43x10^-3 +11x10^-3 ) = 215.8 , but the ans is 112 , what's wrong with my working ?
  38. W

    Centroid of an Area - Finding X & Y Coordinates

    Homework Statement total area = (75x175) +(100x175) +(0.5x50x175) -(π/4 x100x100) = 27964 my centroid for y = (75x175)(87.5) + (100x175)(87.5) +(0.5x150x75)(25) -(π/4 x100x100) (175 - (4(100)/3π) ) / 27964 = 97.13 my centroid for x = (75x175)(37.5) +(100x175) (125) + (0.5x50x175)(225)-(π/4...
  39. G

    Centroid of a composite area

    Homework Statement i have a few question here . 1. the y bar for I should be 16.98, am i right ? 2. The y bar for II should be 50+12+24=86 , am i right ? 3. the x bar for V should be 50 , am i right ? Homework EquationsThe Attempt at a Solution
  40. Yam

    Difference between Centroid and Centre of Pressue

    Homework Statement What is the difference between the Centre of Pressure and Centroid Homework EquationsThe Attempt at a Solution My understanding is that the centre of pressure acts on a centroid. So, how come they can be on different positions for a submerged surface?
  41. S

    Statics: tension in cable holding gate closed

    1. Homework Statement The tension in the cable is 800LBs. Find the depth of water that produces this tension. The gate is hinged at B; the cable is attached at A. (Figure 1)Homework EquationsThe Attempt at a Solution OK, so this is just 2 opposing lever arms: the tension in the cable is...
  42. J-dizzal

    Centroid of a uniform shape, using area

    Homework Statement Find the coordinates of the centroid of the uniform area. Homework Equations equations for centroid coordinates at the top of my paper. The Attempt at a Solution
  43. P

    Finding centroid of a solid

    Homework Statement Use cylindrical coordinates to find the centroid of the solid. The solid that is bounded above by the sphere x2 + y2 + z2 = 2 and below by z = x2 + y2 Homework Equations x = rcos(theta) y= rsin(theta)[/B]The Attempt at a Solution I am having trouble trying to find the...
  44. D

    Find the position of the centroid of the shaded area

    1. Find the position of the centroid of the shaded area http://imgur.com/ieiSPPY 2. The black triangle with the square cut out is where the centroid must be located.I know that the object is symmetrical and the triangle can be divided into parts with two smaller right angled triangles I have...
  45. Alex_Neof

    Centroid position of Lamina

    Homework Statement A lamina is bounded by the x-axis, the y-axis, and the curve ##y = 4 -x^2.## Determine the centroid position ##(\bar{x},\bar{y})## of the lamina. Homework Equations ## A = \int_a^b (f(x) - g(x)) dx ## (Area) ##\bar{x} = \frac{1}{A}\int_a^b x(f(x) - g(x)) dx ## ##\bar{y}...
  46. B

    Finding the centroid of a triangle using complex numbers

    Hi all, I'm preparing for a deferred exam this semester after falling ill last year. Just looking over my course notes and have a question. I understand how this works in the big picture scheme. What I don't understand however is how my instructor simplified the original equation. 1. Homework...
  47. END

    What does ##\bar{x}_{\textrm{el}}## represent?

    In the context of centroids and moments, what do ##\bar{x}_{\textrm{el}}## and ##\bar{y}_{\textrm{el}}## represent? For example: $$\bar{x}L = \int \bar{x}_{\textrm{el}}dL$$
  48. J

    Centroid of Integral Area?

    If I have an integrated area such as the blue area in the link below, what function can be written to find the location on the x-axis where half of the area is one side and half is on the other or more specifically a function that determines the x-axis location of the centroid...
  49. S

    Please help me finding this centroid of a graph. Thank you.

    Homework Statement Find the x-coordinate of the centroid of the region bounded by the graphs of Homework Equations y= 5/(√(25-x2)) The Attempt at a Solution I stuck at finding Mx
  50. P

    Prove that a point in triangle is the centroid

    You are given an arbitrary triangle ABC. Inside ABC there is a point M such that Area(ABM) = Area(BCM) = Area(ACM) . Prove that M is the centroid of triangle ABC. I have had very little progress with this question. I've tried connecting a line from M which bisects BC, but I cannot prove that...