Centroid formula

College/University

Definition

Formula used to find the coordinates of the centroid for a triangle or any sets of points on a X-Y plane. In a triangle, if we average the x and y coordinates of the verticies, it will result in the coordinates of the centroid.

Worked examples

\(\)Triangle with vertices \( A(2, 4), B(6, 8), C(4, 2) \)→\( \)Centroid \( = \left(\frac{2+6+4}{3}, \frac{4+8+2}{3}\right) = (4, \frac{14}{3})\)

Add the three x-coordinates and divide by 3; do the same for the y-coordinates.

\(\)Vertices \( (0, 0), (6, 0), (3, 9) \)→\( \)Centroid \( = \left(\frac{0+6+3}{3}, \frac{0+0+9}{3}\right) = (3, 3)\)

The centroid is the average of all three vertices, giving the triangle's balance point.

Common mistakes

\(\)Centroid\( = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\) → \(\)Centroid\( = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\) A triangle has three vertices, so divide by 3, not 2.
\(\)Centroid of \( (1,2), (3,4), (5,6) = \left(\frac{1+3+5}{2}, \frac{2+4+6}{2}\right) = (4.5, 6)\) → \(\left(\frac{1+3+5}{3}, \frac{2+4+6}{3}\right) = (3, 4)\) Divide the sum by the number of points (3), not by 2.
\(\)Centroid\( = \left(\frac{x_1 \cdot x_2 \cdot x_3}{3}, \frac{y_1 \cdot y_2 \cdot y_3}{3}\right)\) → \(\)Centroid\( = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\) Add the coordinates, don't multiply them.

Where you'll use it next

You'll apply the centroid formula in coordinate geometry when working with medians of triangles, center-of-mass problems in physics, and later in calculus when finding centroids of more complex regions.

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.

UniversityUniversity