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 Math Help > Geometry > Solids > Centroid

The Centroid of a figure is its "balancing point" or center of mass.  A plane figure would "balance" on the tip of a pin stuck in the centroid.  A solid (3-dimensional) figure would balathe tip of a pin stuck in the centroid.  A solid (3-dimensional) figure would balance (i.e. not tend to rotate under the influence of gravity) if it were suspended from its centroid.

## Centroid

To visualize the meaning of the centroid, think of a see-saw.  If two people, one weighing 25 kg and the other weighing 50 kg want to balance on the see-saw, the heavier one will need to sit only half as far from the balancing point as the lighter one.  That is, the contribution of each person to the turning force, or moment, of the see-saw is his mass times his distance from the center of the see-saw.  The centroid is the place to put the center of the see-saw to make it balance.

The moment (turning force) is the sum of the masses times their distance from the center:

M = m1d1  +  m2d2  +  ...  +  mndn

Suppose we wanted to exactly balance this moment using a mass

m1 + m2+ ... + mn

at distance d from the center.  In that case, since M=md, this distance is d=M/m,

 d = m1d1 + m2d2 + ... + mndn    ——————————————— m1 + m2+ ... + mn

Let's denote x coordinate of the centroid X, and the y coordinate y.  From the discussion thus far, it is clear that the centroid X of a finite number of point masses is the sum of the product of each mass and its x-coordinate divided by the sum of all the masses.

When the masses are divided more and more finely, so that each mass is infinitesimally small, and there are infinitely many of them, we turn these sums into the corresponding integral.  (See the Definition of Integral for more about this.)  The centroid X of a plane figure is sum of all the infinitesimally thin slices of the figure times x, their distance along the x-axis, divided by the total area of the figure (which is the integral of their slices, but without multiplying each slice by x).  For example, the area of a quarter circle of radius r is (1/4) π r², so the centroid of a quarter circle given by y=sqrt(r²-x²) is

 4/(π r²) óôõ r 0 x sqrt(r²-x²) dx  = 4/(π r²)(1/3) π r³ = (4/3)(r/π)

### Related pages in this website

Solid of Rotation describes how to calculate it using integrals, and better yet, how to calculate it using Pappus' Centroid Theorem.  It also has a table of surface areas and volumes of a variety of simple solid shapes.

Platonic and Archimedean Solids

Tetrahedron

Solid figures

physics SI units, for a very brief mention of "moment"

The webmaster and author of this Math Help site is Graeme McRae.