Centroid

The geometric center of a region or mass distribution, found by balancing its first moments. For a uniform-density body, it coincides with the center of mass.

First moment of area: The integral of position weighted by area, used to locate a plane region’s centroid. Its ratios to total area give the centroid’s coordinates.
Cartesian coordinate system: A coordinate system that locates points using perpendicular axes and numerical coordinates. Centroid coordinates depend on the chosen origin and axes.
Statics: The branch of mechanics that studies forces and moments in systems at rest or in equilibrium. Centroid locations help reduce distributed loads and analyze equilibrium.
Incenter: The center of a triangle’s inscribed circle, equidistant from its three sides. It is defined by distances to sides, not by first moments of area.
Center of mass: The point representing the mean position of mass in a physical system. It matches the centroid for uniform density, but shifts when density varies.
Integral: A mathematical operation that accumulates contributions over a continuous interval or region. Continuous centroid formulas sum position-weighted contributions through integration.
Shear center: The point on a beam cross-section through which a transverse load causes bending without twisting. Its location is calculated using section geometry and centroid-related properties.
Circumcenter: The center of a circle passing through all vertices of a triangle. It depends on vertex geometry rather than the triangle’s area distribution.
Center of gravity: The point through which the resultant gravitational force on a body acts. In a uniform gravitational field, it coincides with the center of mass and often the centroid.
Geometric symmetry: An invariance of a shape under transformations such as reflection or rotation. Symmetry often fixes a centroid’s location without evaluating integrals.

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