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How to Pinpoint the Center of a Triangle

To find the center of a triangle, all you need are the locations of the three corners and the midpoints of the sides opposite those vertices. (It’s also a good idea to draw the triangle to help you see what you’re doing.)

If you draw lines from each corner (or vertex) of a triangle to the midpoint of the opposite sides, then those three lines meet at the center, or centroid, of the triangle. This point is the triangle’s center of gravity, where the triangle balances evenly. The coordinates of the centroid are also two-thirds of the way from each vertex along that segment. The following figure shows how the three lines drawn in the triangle all meet at the center.

The lines that intersect at the centroid of a triangle.
The lines that intersect at the centroid of a triangle.

To find the centroid of a triangle, use the formula

image1.png

where are the coordinates of the vertex and are the coordinates of the midpoint of the side opposite the vertes. The formula locates a point two-thirds of the distance from the vertex to the midpoint of the opposite side.

For example, to find the centroid of a triangle with vertices at (0,0), (12,0), and (3,9), first find the midpoint of one of the sides. The most convenient side is the bottom, because it lies along the x-axis. The coordinates of that midpoint are (6,0). Then find the point that sits two-thirds of the way from the opposite vertex, (3,9):

  1. Replace x1, x2, y1, and y2 with their respective values; replace k with 2/3.

    image2.png
  2. Simplify the computation to get the point.

    image3.png

In this example, the centroid is the point (5,3), as shown in the following figure.

The graph of a triangle with a centroid at (5,3).
The graph of a triangle with a centroid at (5,3).
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