Triangle Centroid CalculatorFree online math calculator — instant results, no sign-up
Calculate a triangle's centroid from its three vertices.
About the Triangle Centroid Calculator
The Triangle Centroid Calculator finds the centroid of a triangle from the x- and y-coordinates of its three vertices. It averages the three x-coordinates and the three y-coordinates to locate the intersection of the triangle’s medians. With the default vertices (0, 0), (6, 0), and (0, 6), the centroid is (2, 2).
- First x
- The x-coordinate of the first vertex, defaulting to 0, expressed in the same coordinate units as the other entries.
- First y
- The y-coordinate of the first vertex, defaulting to 0, expressed in the same coordinate units as the other entries.
- Second x
- The x-coordinate of the second vertex, defaulting to 6, expressed in the same coordinate units as the other entries.
- Second y
- The y-coordinate of the second vertex, defaulting to 0, expressed in the same coordinate units as the other entries.
- Third x
- The x-coordinate of the third vertex, defaulting to 0, expressed in the same coordinate units as the other entries.
- Third y
- The y-coordinate of the third vertex, defaulting to 6, expressed in the same coordinate units as the other entries.
Formula and method
G = (x_G, y_G), where x_G = (x_1 + x_2 + x_3) / 3 and y_G = (y_1 + y_2 + y_3) / 3. With the default inputs: x_G = (0 + 6 + 0) / 3 = 2; y_G = (0 + 0 + 6) / 3 = 2.
- GThe triangle’s centroid, expressed as an ordered coordinate pair.
- x_GThe x-coordinate of the centroid.
- y_GThe y-coordinate of the centroid.
- x_1, y_1The x- and y-coordinates of the first vertex.
- x_2, y_2The x- and y-coordinates of the second vertex.
- x_3, y_3The x- and y-coordinates of the third vertex.
The calculator treats the inputs as three vertices in a two-dimensional Cartesian coordinate system. It adds the x-coordinates and divides by three, then independently adds the y-coordinates and divides by three. These two averages form the centroid; no side lengths, angles, or area calculations are needed.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- First x0
- First y0
- Second x6
- Second y0
- Third x0
- Third y6
Results
- Centroid x2
- Centroid y2
How to use the Triangle Centroid Calculator
- 1Enter the first vertex in First x and First y; the defaults are 0 and 0.
- 2Enter the second vertex in Second x and Second y; the defaults are 6 and 0.
- 3Enter the third vertex in Third x and Third y; the defaults are 0 and 6.
- 4Check that all coordinates use a consistent coordinate system and units, then calculate the centroid.
- 5Read Centroid x and Centroid y together as an ordered pair; the default inputs give (2, 2).
Reading your result
Centroid x is the horizontal coordinate of the centroid, and Centroid y is its vertical coordinate, both in the same units as the inputs. For the defaults, Centroid x = 2 and Centroid y = 2 place the centroid at (2, 2). For a nondegenerate triangle, this point lies inside the triangle and divides each median in a 2:1 ratio, with the longer portion between the vertex and the centroid.
Limitations
The formulas assume planar Cartesian coordinates, not latitude and longitude on a curved surface. Collinear or repeated vertices do not form a nondegenerate triangle: their coordinate averages still exist, but there is no positive-area triangle. The centroid is the balance point of a uniform-density triangular plate, not necessarily the center of mass of one with uneven density. Decimal inputs and displayed results may be subject to numerical rounding.
Common questions
Does changing the order of the triangle’s vertices change the centroid?+
No. Reordering the three vertices leaves both coordinate sums unchanged, so the centroid stays the same. Keep each vertex’s x- and y-coordinates paired correctly.
Why do the default vertices give a centroid of (2, 2)?+
For (0, 0), (6, 0), and (0, 6), the x-coordinates sum to 6 and the y-coordinates also sum to 6. Dividing each sum by the three vertices gives Centroid x = 2 and Centroid y = 2.
Is the centroid always inside the triangle?+
Yes, for any nondegenerate triangle, the centroid lies strictly inside it. If the vertices are collinear or repeated, the triangle has zero area and this interior-point statement does not apply.
Is the centroid the same as the circumcenter or incenter?+
Not generally. The centroid is where the medians meet, the circumcenter is equidistant from the vertices, and the incenter is equidistant from the sides. All three coincide in an equilateral triangle.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
You might also like
Slope Calculator
Slope, angle and line equation through two points.
Area Calculator
Area of common 2D shapes.
Volume Calculator
Volume of common 3D shapes.
Surface Area Calculator
Surface area of 3D shapes.
Circle Calculator
Radius, diameter, circumference and area.
Triangle Calculator
Solve a triangle from three sides.
Results are estimates for informational purposes and are not financial, medical or legal advice.