Key takeaway: The centroid is the point where the three medians of a triangle intersect, located two thirds of the way from each vertex to the midpoint of the opposite side.
Grade 10 – GeometryG-CO.C.10Geometry and Measurement
Definition of Centroid
The centroid is the balance point of a triangle. A cardboard triangle rests level on a pencil tip placed exactly there. It is where the three medians cross, and a median runs from a corner to the midpoint of the opposite side.
This point always lies inside the shape. On every median it cuts the segment in a 2 to 1 ratio, with the longer part next to the corner. On a coordinate grid, you find it by averaging the corner coordinates.
Formula and notation
For corners \((p, q)\), \((r, s)\) and \((t, u)\), the centroid \(G\) is:
\[ G = ( \frac{p + r + t}{3},\ \frac{q + s + u}{3} ) \]
Along a median from corner \(A\) to midpoint \(M\): \(AG = \frac{2}{3}\,AM\) and \(GM = \frac{1}{3}\,AM\).
Key facts
- All three medians pass through one shared point.
- The piece next to the corner is twice as long as the piece next to the midpoint.
- Acute, right and obtuse triangles all keep this point inside.
- A triangular plate of even thickness balances exactly at this spot.
Centroid example
A triangular garden bed has corners at \(A(2, 5)\), \(B(8, -1)\) and \(C(-1, 2)\), in feet on a grid. Find where a single post must go to balance the bed, and how far that post is from \(A\) along the median if the whole median to \(M\) is 15 ft long.
-
Average the first coordinates: \(\frac{2 + 8 + (-1)}{3} = \frac{9}{3} = 3\).
-
Average the second coordinates: \(\frac{5 + (-1) + 2}{3} = \frac{6}{3} = 2\), so \(G = (3, 2)\).
-
Distance from the corner: \(AG = \frac{2}{3}(15) = 10\) ft.
The post goes at \((3, 2)\), which is 10 ft from \(A\) along that median.
Common mistake
Watch out
Pia says the balance point is halfway along each median, so \(AG = 7.5\) ft on the 15-ft median.
Why it is wrong
Halfway is the midpoint \(M\) of the side, not the balance point. The correct spot is two thirds of the way from the corner, so \(AG = 10\) ft and \(GM = 5\) ft. Check: the longer piece is twice the shorter one.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement