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

$$M = \int\!\!\!\int \sigma(x,y)\,dA.$$ (1)

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

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

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

$$M = \int\!\!\!\int\!\!\!\int \rho(x,y,z)\,dV,$$ (4)

and the coordinates of the center of mass are

$${\bar x} = {\int\!\!\!\int\!\!\!\int x\rho(x,y,z)\,dV\over M}$$ (5)

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

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

References

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.