# List of centroids

The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object ${\displaystyle X}$ in ${\displaystyle n}$-dimensional space is the intersection of all hyperplanes that divide ${\displaystyle X}$ into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of ${\displaystyle X}$. 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 ${\displaystyle ({\bar {x}},{\bar {y}})}$ are given:

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

The area between the curve ${\displaystyle y={\frac {h}{b^{2}}}x^{2}}$ and the ${\displaystyle \,\!y}$ axis, from ${\displaystyle \,\!y=0}$ to ${\displaystyle \,\!y=h}$

${\displaystyle {\frac {3b}{8}}}$ ${\displaystyle {\frac {3h}{5}}}$ ${\displaystyle {\frac {2bh}{3}}}$
Parabolic spandrel The area between the curve ${\displaystyle \,\!y={\frac {h}{b^{2}}}x^{2}}$ and the ${\displaystyle \,\!x}$ axis, from ${\displaystyle \,\!x=0}$ to ${\displaystyle \,\!x=b}$ ${\displaystyle {\frac {3b}{4}}}$ ${\displaystyle {\frac {3h}{10}}}$ ${\displaystyle {\frac {bh}{3}}}$
General spandrel The area between the curve ${\displaystyle y={\frac {h}{b^{n}}}x^{n}}$ and the ${\displaystyle \,\!x}$ axis, from ${\displaystyle \,\!x=0}$ to ${\displaystyle \,\!x=b}$ ${\displaystyle {\frac {n+1}{n+2}}b}$ ${\displaystyle {\frac {n+1}{4n+2}}h}$ ${\displaystyle {\frac {bh}{n+1}}}$

## 3-D Centroids

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

Shape Figure ${\displaystyle {\bar {x}}}$ ${\displaystyle {\bar {y}}}$ ${\displaystyle {\bar {z}}}$ Volume
Cuboid a, b = the sides of the cuboid's base
c = the third side of the cuboid
${\displaystyle {\frac {a}{2}}}$ ${\displaystyle {\frac {b}{2}}}$ ${\displaystyle {\frac {c}{2}}}$ ${\displaystyle {abc}}$
Right-rectangular pyramid a, b = the sides of the base
h = the distance is from base to the apex
${\displaystyle {\frac {a}{2}}}$ ${\displaystyle {\frac {b}{2}}}$ ${\displaystyle {\frac {h}{4}}}$ ${\displaystyle {\frac {abh}{3}}}$
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
${\displaystyle {\frac {h}{3}}}$ ${\displaystyle {\frac {L}{2}}}$ ${\displaystyle {\frac {bhL}{2}}}$
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
${\displaystyle {\frac {b}{2}}}$ ${\displaystyle {\frac {h}{3}}}$ ${\displaystyle {\frac {L}{2}}}$ ${\displaystyle {\frac {bhL}{2}}}$
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
${\displaystyle {\frac {b}{3}}}$ ${\displaystyle {\frac {h}{3}}}$ ${\displaystyle {\frac {L}{2}}}$ ${\displaystyle {\frac {bhL}{2}}}$
Right circular cylinder r = the radius of the cylinder
h = the height of the cylinder
${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {h}{2}}}$ ${\displaystyle {\pi r^{2}h}}$
Right circular solid cone r = the radius of the cone's base
h = the distance is from base to the apex
${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {h}{4}}}$ ${\displaystyle {\frac {\pi r^{2}h}{3}}}$
Solid sphere r = the radius of the sphere ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {4\pi r^{3}}{3}}}$
Solid hemisphere r = the radius of the hemisphere ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {3r}{8}}}$ ${\displaystyle {\frac {2\pi r^{3}}{3}}}$
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
${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {3h}{8}}}$ ${\displaystyle {\frac {2\pi a^{2}h}{3}}}$
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
${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {h}{3}}}$ ${\displaystyle {\frac {\pi a^{2}h}{2}}}$
Solid ellipsoid a, b, c = the principal semi-axes of the ellipsoid ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {4\pi abc}{3}}}$
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
${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {3c}{8}}}$ ${\displaystyle {\frac {2\pi abc}{3}}}$
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
${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {c}{3}}}$ ${\displaystyle {\frac {\pi abc}{2}}}$