Centroid

College/University

Definition

One of the 4 main "centers of a traingle." Centroid refers to the center of mass of a figure. In a triangle, the centroid is located by drawing 3 medians (lines from the vertex to the midpoint of the opposite side). The point where the three lines intersect is the centroid of the triangle.

Worked examples

\(\triangle ABC\): draw medians from \(A\), \(B\), \(C\) to midpoints of opposite sides; they meet at point \(G\).

Point \(G\) is the centroid — the center of mass where the triangle would balance on a pin.

If \(G\) is the centroid and \(M\) is a midpoint, then \(AG = \frac{2}{3} AM\).

The centroid divides each median in a 2:1 ratio, closer to the vertex than the midpoint.

Common mistakes

The centroid is at the midpoint of each median → The centroid divides each median in a 2:1 ratio from vertex to midpoint The centroid is ⅔ of the way from any vertex along its median, not halfway.
Drawing altitudes (perpendicular heights) locates the centroid → Draw medians (vertex to opposite midpoint) to find the centroid Altitudes meet at the orthocenter, not the centroid. Medians find the centroid.
The centroid is the same as the circumcenter or incenter → Centroid, circumcenter, incenter, and orthocenter are four distinct triangle centers They coincide only in an equilateral triangle; otherwise each has a unique construction and location.

Where you'll use it next

You'll use the centroid when studying triangle centers, solving coordinate geometry problems to find balance points, and later in physics for center-of-mass calculations of composite shapes.

Found in 1 StudyPug lesson

Mastering Moment and Center of Mass in Calculus 3

Multivariable Calculus

Dive deep into moment and center of mass concepts, essential for advanced physics and engineering. Learn to calculate, analyze, and apply these principles to real-world problems in Calculus 3.

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