Centroid - Triangle Centroid From Coordinates

A centroid calculator finds the geometric center of a triangle from three vertex coordinates. The centroid is the point where the three medians meet, and it equals the arithmetic mean of the three vertices: Cx = (x1 + x2 + x3) / 3 and Cy = (y1 + y2 + y3) / 3. It is the most common way to summarize a triangle with a single point.

• • Coordinate geometry homework: Verify textbook exercises where the triangle is given as three coordinate points and the centroid must be reported in the same length unit as the input grid.
• • Plotting a triangle center: Mark the centroid of a survey or sketch triangle on a map or drawing without recomputing the medians by hand.
• • Pre-step for area or moment calculations: Use the centroid as the reference point for further geometric work such as triangle area via the shoelace formula or for second-moment calculations.
• • Graphics and CAD triangle centers: Compute the centroid of a triangle stored in a model as three vertex coordinates, especially when the triangle is not axis-aligned and a base-height shortcut does not apply.

A triangle in the plane is uniquely defined by three non-collinear points, so its centroid is a function of six coordinates. The arithmetic mean of the three vertices collapses those numbers into a single point, and the medians cross-check confirms the result geometrically.

When the input is given as side lengths or a base and a height instead of three vertex coordinates, the Triangle Calculator handles the SSS, SAS, and base-height cases in the same Math and Conversion cluster.

The centroid calculator averages the three x-coordinates to get Cx, averages the three y-coordinates to get Cy, and reports the centroid as a single point (Cx, Cy) in the same length unit as the inputs.

• x1, y1: Cartesian coordinates of the first vertex of the triangle, in any length unit.
• x2, y2: Cartesian coordinates of the second vertex of the triangle, in the same length unit as vertex 1.
• x3, y3: Cartesian coordinates of the third vertex of the triangle, in the same length unit as the other two vertices.
• Cx: x-coordinate of the centroid, equal to (x1 + x2 + x3) / 3, in the same length unit as the input x values.
• Cy: y-coordinate of the centroid, equal to (y1 + y2 + y3) / 3, in the same length unit as the input y values.

The arithmetic mean of the three vertices is the simplest form of the centroid formula for a triangle, and it returns the same point as the intersection of the medians. The calculator also reports the centroid as a formatted (Cx, Cy) pair for direct plotting.

The output is in the same length unit as the input coordinates. Meters in gives meters out, feet in gives feet out, and inches in gives inches out. The calculator does not convert units, so use a separate area or length converter if the report needs a different unit.

According to Wolfram MathWorld, the centroid of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is the point ((x1+x2+x3)/3, (y1+y2+y3)/3), which is the intersection of the three medians and the geometric center of the triangle.

The same six coordinates that produce the centroid also produce the triangle area through the shoelace formula, so the Area Triangle Coordinates Calculator is the natural next step after this centroid calculator.

Four ideas decide what the centroid means for a triangle and how the formula behaves on edge cases.

The output is in the same length unit as the input coordinates. Meters in gives meters out, feet in gives feet out, and inches in gives inches out. The calculator does not convert units, so use a length converter if the report needs a different unit.

When the three vertices carry different masses, the geometric centroid generalizes to the mass-weighted center on the Center Of Mass Calculator in the same Math and Conversion cluster.

Use the centroid calculator with all six coordinates in the same length unit, then read the centroid (Cx, Cy) and the medians cross-check in the same pass.

1. 1 Enter the first vertex: Type the x and y coordinates of the first vertex. Use the same length unit for both coordinates and for the other two vertices.
2. 2 Enter the second vertex: Type the x and y coordinates of the second vertex. The order of the vertices does not change the centroid because the formula uses the arithmetic mean.
3. 3 Enter the third vertex: Type the x and y coordinates of the third vertex. The results panel updates the centroid (Cx, Cy) and the medians cross-check in real time as you type.
4. 4 Read the centroid: Use the centroid point in the black results panel as the primary answer. Cx and Cy are also listed below as separate numeric rows in the same length unit as the inputs.
5. 5 Cross-check with the medians: The page shows the medians cross-check, which is the centroid computed again from the midpoints of the three sides. The two values match for any non-degenerate triangle, which is a quick way to confirm the inputs were entered correctly.

For a triangle with vertices (0, 0), (6, 0), and (0, 9), the centroid is (2, 3). The result panel reports Cx = 2 and Cy = 3, and the medians cross-check matches the same point.

For a right triangle with the two legs along the axes, the centroid lands at one third of each leg, which is a useful shortcut the Right Triangle Calculator explores in more depth.

The arithmetic-mean formula is short, but the calculator wraps it in a layout that helps with plotting, verifying, and reporting the centroid of a triangle.

• • Direct from six coordinates: The arithmetic-mean formula takes the three vertices straight to the centroid, so no medians, side lengths, or perpendicular heights are needed first.
• • Centroid point is shown explicitly: The page formats the centroid as (Cx, Cy) so it can be plotted on a coordinate plane or pasted into a report without further formatting.
• • Medians cross-check is included: The cross-check recomputes the centroid from the midpoints of the three sides, so a single page can confirm the arithmetic mean and the median intersection at the same time.
• • Negative and large coordinates are supported: Decimal coordinates, negative coordinates, and coordinates far from the origin all work, as long as all six values are in the same length unit.
• • Edge cases are flagged: Collinear and identical vertices are kept as inputs but the page notes that the geometric centroid is not defined for a degenerate triangle.

The calculator returns the centroid point, the two coordinate rows, and the medians cross-check together, so the right value for the next step is always on the page.

The same three vertex coordinates that feed this centroid calculator also feed the Triangle Area Calculator, which is the natural pair when both the center and the area of a triangle are needed in the same report.

The arithmetic-mean formula is stable, but a few input choices decide whether the page returns the right value for a given triangle.

• • The calculator does not solve for a missing vertex when only the centroid and two vertices are known. Use a system of equations for that case instead.
• • Hand calculation that rounds the mean partway through can differ by a few hundredths of a length unit from the calculator, which keeps full precision until the display step.

The arithmetic-mean formula generalizes to polygons with more sides through the shoelace-based polygon centroid formula, which is the natural next step for shapes beyond the triangle.

According to Wikipedia, the centroid of a triangle is constructed as the common intersection of the three medians, and for a triangle with vertices (x1,y1), (x2,y2), (x3,y3) the centroid equals the arithmetic mean of the three vertices.

The arithmetic-mean formula generalizes to polygons with more sides through the shoelace-based polygon centroid formula, which the Polygon Area Calculator applies to general n-vertex shapes in the same Math and Conversion cluster.