List of centroids

The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object $X$ in $n$ -dimensional space is the intersection of all hyperplanes that divide $X$ into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of $X$ . For an object of uniform composition, the centroid of a body is also its center of mass. In the case of two-dimensional objects shown below, the hyperplanes are simply lines.

2-D Centroids

For each two-dimensional shape below, the area and the centroid coordinates $({\bar {x}},{\bar {y}})$ are given:

Shape	Figure	${\bar {x}}$	${\bar {y}}$	Area
rectangle area		0	0	${bh}$
General triangular area		${\frac {x1+x2+x3}{3}}$ ^[1]	${\frac {h}{3}}$	${\frac {bh}{2}}$
Isosceles-triangular area		${\frac {l}{2}}$	${\frac {h}{3}}$	${\frac {lh}{2}}$
Right-triangular area		${\frac {b}{3}}$	${\frac {h}{3}}$	${\frac {bh}{2}}$
Circular area		$0$	$0$	${\pi r^{2}}$
Quarter-circular area^[2]		${\frac {4r}{3\pi }}$	${\frac {4r}{3\pi }}$	${\frac {\pi r^{2}}{4}}$
Semicircular area^[3]		$0$	${\frac {4r}{3\pi }}$	${\frac {\pi r^{2}}{2}}$
Circular sector		${\frac {2r\sin(\alpha )}{3\alpha }}$	$\,\!0$	$\,\!\alpha r^{2}$
Circular segment		${\frac {4r\sin ^{3}(\alpha )}{3(2\alpha -\sin(2\alpha ))}}$	$\,\!0$	${\frac {r^{2}}{2}}(2\alpha -\sin(2\alpha ))$
Quarter-circular arc	The points on the circle $\,\!x^{2}+y^{2}=r^{2}$ and in the first quadrant	${\frac {2r}{\pi }}$	${\frac {2r}{\pi }}$	$L={\frac {\pi r}{2}}$
Semicircular arc	The points on the circle $\,\!x^{2}+y^{2}=r^{2}$ and above the $\,\!x$ axis	$\,\!0$	${\frac {2r}{\pi }}$	$L=\,\!\pi r$
Arc of circle	The points on the curve (in polar coordinates) $\,\!\rho =r$ , from $\,\!\theta =-\alpha$ to $\,\!\theta =\alpha$	${\frac {\rho \sin(\alpha )}{\alpha }}$	$\,\!0$	$L=\,\!2\alpha \rho$
elliptical area		$0$	$0$	${\pi ab}$
Quarter-elliptical area		${\frac {4a}{3\pi }}$	${\frac {4b}{3\pi }}$	${\frac {\pi ab}{4}}$
Semielliptical area		$\,\!0$	${\frac {4b}{3\pi }}$	${\frac {\pi ab}{2}}$
Parabolic area	The area between the curve $\,\!y={\frac {h}{b^{2}}}x^{2}$ and the line $\,\!y=h$	$\,\!0$	${\frac {3h}{5}}$	${\frac {4bh}{3}}$
Semiparabolic area The area between the curve $y={\frac {h}{b^{2}}}x^{2}$ and the $\,\!y$ axis, from $\,\!y=0$ to $\,\!y=h$		${\frac {3b}{8}}$	${\frac {3h}{5}}$	${\frac {2bh}{3}}$
Parabolic spandrel	The area between the curve $\,\!y={\frac {h}{b^{2}}}x^{2}$ and the $\,\!x$ axis, from $\,\!x=0$ to $\,\!x=b$	${\frac {3b}{4}}$	${\frac {3h}{10}}$	${\frac {bh}{3}}$
General spandrel	The area between the curve $y={\frac {h}{b^{n}}}x^{n}$ and the $\,\!x$ axis, from $\,\!x=0$ to $\,\!x=b$	${\frac {n+1}{n+2}}b$	${\frac {n+1}{4n+2}}h$	${\frac {bh}{n+1}}$

3-D Centroids

For each three-dimensional body below, the volume and the centroid coordinates $({\bar {x}},{\bar {y}})$ , $({\bar {z}})$ are given:

Shape	Figure	${\bar {x}}$	${\bar {y}}$	${\bar {z}}$	Volume
Cuboid	a, b = the sides of the cuboid's base c = the third side of the cuboid	${\frac {a}{2}}$	${\frac {b}{2}}$	${\frac {c}{2}}$	${abc}$
Right-rectangular pyramid	a, b = the sides of the base h = the distance is from base to the apex	${\frac {a}{2}}$	${\frac {b}{2}}$	${\frac {h}{4}}$	${\frac {abh}{3}}$
General triangular prism	b = the base side of the prism's triangular base, h = the height of the prism's triangular base L = the length of the prism	see above for general triangular base	${\frac {h}{3}}$	${\frac {L}{2}}$	${\frac {bhL}{2}}$
Isoscele triangular prism	b = the base side of the prism's triangular base, h = the height of the prism's triangular base L = the length of the prism	${\frac {b}{2}}$	${\frac {h}{3}}$	${\frac {L}{2}}$	${\frac {bhL}{2}}$
Right-triangular prism	b = the base side of the prism's triangular base, h = the perpendicular side of the prism's triangular base L = the length of the prism	${\frac {b}{3}}$	${\frac {h}{3}}$	${\frac {L}{2}}$	${\frac {bhL}{2}}$
Right circular cylinder	r = the radius of the cylinder h = the height of the cylinder	$0$	$0$	${\frac {h}{2}}$	${\pi r^{2}h}$
Right circular solid cone	r = the radius of the cone's base h = the distance is from base to the apex	$0$	$0$	${\frac {h}{4}}$	${\frac {\pi r^{2}h}{3}}$
Solid sphere	r = the radius of the sphere	$0$	$0$	$0$	${\frac {4\pi r^{3}}{3}}$
Solid hemisphere	r = the radius of the hemisphere	$0$	$0$	${\frac {3r}{8}}$	${\frac {2\pi r^{3}}{3}}$
Solid semi-ellipsoid of revolution around z-axe	a = the radius of the base circle h = the height of the semi-ellipsoid from the base cicle's center to the edge	$0$	$0$	${\frac {3h}{8}}$	${\frac {2\pi a^{2}h}{3}}$
Solid paraboloid of revolution around z-axe	a = the radius of the base circle h = the height of the paboloid from the base cicle's center to the edge	$0$	$0$	${\frac {h}{3}}$	${\frac {\pi a^{2}h}{2}}$
Solid ellipsoid	a, b, c = the principal semi-axes of the ellipsoid	$0$	$0$	$0$	${\frac {4\pi abc}{3}}$
Solid semi-ellipsoid around z-axe	a, b = the principal semi-axes of the base ellipse c = the principal z-semi-axe from the center of base ellipse	$0$	$0$	${\frac {3c}{8}}$	${\frac {2\pi abc}{3}}$
Solid paraboloid around z-axe	a, b = the principal semi-axes of the base ellipse c = the principal z-semi-axe from the center of base ellipse	$0$	$0$	${\frac {c}{3}}$	${\frac {\pi abc}{2}}$