Centroid
In short: A triangle’s centroid is its balance point: the single spot where a cardboard cutout of the triangle would rest level on a pencil tip, found by crossing the three medians.
GeometryGeometryG-CO.C.10
What is Centroid?
Cut a triangle from cardboard and balance it on a pencil tip: the one spot that works is the centroid. Geometrically, you reach it by joining each vertex to the midpoint across from it; those three joining lines, the medians, always cross at a single point.
That crossing point splits every median 2 to 1, the longer share touching the vertex. For vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), simply average each coordinate: \(\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\). The result is never outside the triangle.
How to spot it
- A problem asks where a triangular sign, plate or sail would balance on a single support.
- The statement mentions medians, or says three lines drawn to midpoints are concurrent.
- Three corner coordinates are listed and you need their mean position.
- One piece of a median is known, and you must find another piece using a 2 to 1 split.
The method
- With coordinates, add the three x-values and divide by 3; then do likewise for the y-values.
- With lengths, take two-thirds of the whole median to reach the point from the vertex.
- The leftover third runs from the point to the midpoint, exactly half as long as the other share.
- Sanity check: the answer must lie inside the triangle, because a balance point cannot hang outside.
Centroid: solved example
Problem
A triangular metal plate has corners at \(A(1, 2)\), \(B(7, -1)\) and \(C(4, 8)\), with units in inches. Find the point where the plate would balance on a pin.
Solution
Average the x-values: \(\frac{1 + 7 + 4}{3} = \frac{12}{3} = 4\). Average the y-values: \(\frac{2 + (-1) + 8}{3} = \frac{9}{3} = 3\).
Verify along one median: the midpoint of \(\overline{BC}\) is \((5.5, 3.5)\). From \(A\), two-thirds of the offset \((4.5, 1.5)\) is \((3, 1)\), and \((1, 2) + (3, 1) = (4, 3)\). The balance point is \((4, 3)\).
Try it
Question 1
Find the balance point of the triangle whose corners are \((-2, 5)\), \((10, 1)\) and \((4, 15)\).
Show the answer
\(\left(\frac{-2 + 10 + 4}{3}, \frac{5 + 1 + 15}{3}\right) = (4, 7)\).
Question 2
One median of a triangular roof panel measures 27 inches. How far from its vertex does the balance point sit?
Show the answer
Two-thirds of the median: \(\frac{2}{3} \times 27 = 18\) inches, leaving 9 inches to the midpoint.
Centroid vs. Incenter
The centroid is the mean position of the vertices and answers a balance question. The incenter is where the angle bisectors cross and answers a different question: where does the biggest inside circle sit? In a long, skinny triangle the two points land in visibly different places.
Keep going
Lessons
Worksheets and quizzes
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
- Coordinate Geometry and Proofs – Worksheet
- Coordinate Geometry and Proofs – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry