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Centroid (Geometric)

The Center of Mass of a 2-D planar Lamina or a 3-D solid. The mass of a Lamina with surface density function $\sigma(x,y)$ is

M = \int\!\!\!\int \sigma(x,y)\,dA.
\end{displaymath} (1)

The coordinates of the centroid (also called the Center of Gravity) are

{\bar x} = {\int\!\!\!\int x\sigma(x,y)\,dA\over M}
\end{displaymath} (2)

{\bar y} = {\int\!\!\!\int y\sigma(x,y)\,dA\over M}.
\end{displaymath} (3)

The centroids of several common laminas along the nonsymmetrical axis are summarized in the following table.

Figure $\bar y$
Parabolic Segment ${\textstyle{3\over 5}}h$
Semicircle ${4r\over 3\pi}$

In 3-D, the mass of a solid with density function $\rho(x,y,z)$ is

M = \int\!\!\!\int\!\!\!\int \rho(x,y,z)\,dV,
\end{displaymath} (4)

and the coordinates of the center of mass are
{\bar x} = {\int\!\!\!\int\!\!\!\int x\rho(x,y,z)\,dV\over M}
\end{displaymath} (5)

{\bar y} = {\int\!\!\!\int\!\!\!\int y\rho(x,y,z)\,dV\over M}
\end{displaymath} (6)

{\bar z} = {\int\!\!\!\int\!\!\!\int z\rho(x,y,z)\,dV\over M}.
\end{displaymath} (7)

Figure $\bar z$
Cone ${\textstyle{1\over 4}}h$
Conical Frustum ${h({R_1}^2+2R_1R_2+3{R_2}^2)\over 4({R_1}^2+R_1R_2+{R_2}^2)}$
Hemisphere ${\textstyle{3\over 8}}R$
Paraboloid ${\textstyle{2\over 3}}h$
Pyramid ${\textstyle{1\over 4}}h$

See also Pappus's Centroid Theorem


Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987.

McLean, W. G. and Nelson, E. W. ``First Moments and Centroids.'' Ch. 9 in Schaum's Outline of Theory and Problems of Engineering Mechanics: Statics and Dynamics, 4th ed. New York: McGraw-Hill, pp. 134-162, 1988.

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© 1996-9 Eric W. Weisstein