Library / Geometry, coordinates, surveying and trigonometry / GEO-15
Geometry, coordinates, surveying and trigonometry
Centroid of a triangle from its vertices
The x-coordinate of the point where the medians meet, as the average of the three vertex x-coordinates.
- Ref
- GEO-15
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- Gx [cm]
The centroid of a triangle is where its medians meet, and is the average of the three vertices. Its x-coordinate is (x₁ + x₂ + x₃)/3 and its y-coordinate is (y₁ + y₂ + y₃)/3, which is shown in the derived rows. A uniform triangular plate balances on this point. The y-values do not change G_x.
The x-coordinate of the point where the medians meet, as the average of the three vertex x-coordinates. The same rule with the y-values gives G_y, shown in the derived rows. Coordinates depend on the chosen origin.
Calculator
- Centroid y-coordinate, (y₁ + y₂ + y₃)/31 cm
- Distance of the centroid from the origin2.236 cm
- Triangle area (to check the vertices are not collinear)9 cm²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The vertices have an average x of 2 cm and an average y of 1 cm, so the plate balances at (2, 1).
- The centroid sits inside the triangle for any shape; shifting the origin shifts all the coordinates by the same amount.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| x2 Vertex 2, x | −10 to 10 cm | −3.333 | 3.333 | 6.667 | +0.4 cm | +6.67 % |
| x1 Vertex 1, x | −10 to 10 cm | −1.333 | 5.333 | 6.667 | +0.4 cm | +6.67 % |
| x3 Vertex 3, x | −10 to 10 cm | −1.333 | 5.333 | 6.667 | +0.4 cm | +6.67 % |
| y1 Vertex 1, y | −10 to 10 cm | 2 | 2 | 0 | +0.4 cm | 0 % |
| y2 Vertex 2, y | −10 to 10 cm | 2 | 2 | 0 | +0.4 cm | 0 % |
| y3 Vertex 3, y | −10 to 10 cm | 2 | 2 | 0 | +0.4 cm | 0 % |
Sweep of Vertex 2, x
| Vertex 2, x (cm) | Centroid x-coordinate (cm) |
|---|---|
| −10 | −3.333 |
| −7.5 | −2.5 |
| −5 | −1.667 |
| −2.5 | −0.8333 |
| 0 | 0 |
| 2.5 | 0.8333 |
| 5 | 1.667 |
| 7.5 | 2.5 |
| 10 | 3.333 |
Sweep one variable
Variables and how each one contributes
Centroid x-coordinate
What the equation gives you. Everything else feeds it.
Vertex 1, x
The x-coordinate of vertex 1, in cm. Moves the centroid by a third of the change. Raising it by 0.5 (to 0.5) moves the result by +8.33 %, from 2 to 2.167 cm.
Vertex 1, y
The y-coordinate of vertex 1, in cm. It does not change G_x; it sets G_y, shown in the derived rows. Does not change G_x. It moves G_y by a third of any change.
Vertex 2, x
The x-coordinate of vertex 2, in cm. Moves the centroid by a third of the change. Raising it by 0.5 (to 6.5) moves the result by +8.33 %, from 2 to 2.167 cm.
Vertex 2, y
The y-coordinate of vertex 2, in cm. It does not change G_x; it sets G_y. Does not change G_x. It moves G_y by a third of any change.
Vertex 3, x
The x-coordinate of vertex 3, in cm. Moves the centroid by a third of the change. Raising it by 0.5 (to 0.5) moves the result by +8.33 %, from 2 to 2.167 cm.
Vertex 3, y
The y-coordinate of vertex 3, in cm. It does not change G_x; it sets G_y. Does not change G_x. It moves G_y by a third of any change.
Worked example with the values above
- Add the x-values: 0 + 6 + 0 = 6.
- Divide by 3: G_x = 2 cm.
- The y-values give G_y = 1 cm, so G = (2, 1) cm.
Assumptions and limits
- Three vertices in the same Cartesian frame, with the same unit.
- The triangle is a uniform plate if the centroid is used as a balance point.
Common mistakes
- Averaging only two vertices.
- Mixing the units of x and y, or taking vertices from different frames.
- Confusing the centroid with the circumcentre or the incentre, which are different points in general.
Statistics and uncertainty
Each coordinate is an average, so random errors in the vertex positions are reduced to one third of their size.
Where it is used
- Fitting the largest circle in a triangle and the circle through three points.
- Centre lines, balance points and bracing positions of triangular frames and plates.
Maths you need
- Triangle inequality: any two sides add to more than the third
- Heron’s formula: area from three sides
- Midpoint: the point halfway along a segment