Triangle Centroid Calculator
Centroid, area, and perimeter from three corners.
What it does
The centroid is the balance point of a triangle: hang the shape from it and it rests level. Medians from each vertex to the opposite midpoint all cross at this one point, two thirds of the way down each.
The tool above takes three vertex coordinates in any order. It returns the centroid with area and perimeter beside it, so mapping and cutting work share one stop.
The defaults (0,0), (6,0), (0,9) center on (2, 3) with area 27 and perimeter 25.82. The classic (0,0), (0,3), (3,0) centers on (1, 1) with area 4.5.
The average matches OpenStax Calculus centers of mass, whose triangular lamina example lands exactly on the vertex mean. Whitman calculus confirms the same center of mass ground for uniform density.
How to use
- Enter the three corner coordinates.
- Any corner order works.
- Read the centroid point first.
- Read area beside it.
- Read perimeter for edging.
- Balance or cut through the centroid.
Formula
- Cx, Cy
- Centroid coordinates, vertex means.
- area
- Half absolute shoelace sum.
- 2:1
- Median split, vertex side double.
Uniform density assumed; weighted plates need integral centers of mass instead. Zero area from collinear or repeated points has no centroid, and the tool stops those inputs.
Worked example
Example: (0,0), (6,0), (0,9)
Defaults: corners (0,0), (6,0), (0,9).
- Cx: (0 + 6 + 0) / 3 = 2.
- Cy: (0 + 0 + 9) / 3 = 3.
- Area: half of |0 + 54 + 0| = 27.
- Sides: 6, 9, root 117 = 10.82.
- Perimeter: 6 + 9 + 10.82 = 25.82.
- Tool shows (2, 3), 27 sq units, 25.82 units.
Reference table
| Corners | Centroid | Area |
|---|---|---|
| (0,0) (6,0) (0,9) | (2, 3) | 27 sq units |
| (0,0) (0,3) (3,0) | (1, 1) | 4.5 sq units |
| Medians | Cross once | Split 2 to 1 |
Limits
- Uniform density assumed.
- Collinear points rejected.
- Huge coordinates risk float error.
- Order never matters.
Common mistakes
- Averaging only two corners.
- Using midpoints instead of vertices.
- Accepting zero area triangles.
- Reading perimeter where area belongs.
- Expecting centroid at a corner.
Try it yourself
Quick checks. Answers sit tucked under each question.
(0,0), (0,3), (3,0)?
(1, 1) with 4.5 sq units.
Corner order changed?
Same centroid; the mean is symmetric.
Median split?
2 to 1 with the long side at the vertex.
Questions
What is a centroid?
The triangle balance point where all three medians cross.
How is it figured?
Average the three x values and the three y values.
Centroid vs circumcenter?
Centroid balances area; circumcenter centers the circumscribed circle.
Does order matter?
No, vertex means ignore ordering.
Weighted triangle?
Use center of mass integrals for uneven density.
Real world use?
Balancing cutouts, hanging signs, and mapping zones.