Find the centroid, incenter, circumcenter, and orthocenter of a triangle from three vertex coordinates. Computes area, perimeter, side lengths, and angles.
The centroid of a triangle is the point where its three medians intersect. A median connects a vertex to the midpoint of the opposite side, and the centroid divides each median in a 2:1 ratio from the vertex. For a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the centroid is simply ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) — the arithmetic mean of the coordinates.
The centroid is also the center of mass of a uniform triangular plate, which is why physicists and engineers care about it: if you balanced a triangular sheet on a pin at its centroid, it would stay level. It is always located inside the triangle, regardless of the triangle's shape.
Beyond the centroid, every triangle has three other classic centers. The incenter — intersection of angle bisectors — is the center of the inscribed circle (incircle). The circumcenter — intersection of perpendicular bisectors — is the center of the circumscribed circle and may lie outside the triangle for obtuse triangles. The orthocenter — intersection of altitudes — lies inside acute triangles, at the right-angle vertex for right triangles, and outside for obtuse triangles. All four centers are collinear on the Euler line (except the incenter, which is generally off it).
This calculator takes three vertex coordinates and computes all four triangle centers, plus area, perimeter, side lengths, and interior angles. Presets let you explore equilateral, isosceles, right, and scalene cases. A reference table summarises each center's properties.
This centroid of a triangle calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Vertex A — x₁, Vertex A — y₁, Vertex B — x₂, Vertex B — y₂ and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Centroid: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) Incenter: I = (a·x₁+b·x₂+c·x₃)/(a+b+c), same for y (weighted by opposite side lengths) Circumcenter: equidistant from all 3 vertices (perpendicular bisector intersection) Orthocenter: H = 3G − 2O (from the Euler line relation) Area: ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Result: For x1=0, y1=0, x2=6, the tool returns the solved centroid of a triangle outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in centroid of a triangle formulas and reports derived values, checks, and classifications automatically.
This page is tailored to centroid of a triangle, with outputs tied directly to the form fields (Vertex A — x₁, Vertex A — y₁, Vertex B — x₂, Vertex B — y₂). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
The centroid is the intersection of the three medians. It is the center of mass of a uniform triangular plate and is always located inside the triangle at ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
The centroid is the intersection of medians (center of mass), while the incenter is the intersection of angle bisectors (center of the inscribed circle). They coincide only for equilateral triangles.
Yes. For obtuse triangles, the circumcenter lies outside the triangle, on the opposite side of the longest side from the obtuse angle.
The Euler line passes through the centroid, circumcenter, orthocenter, and nine-point center. The centroid divides the segment from circumcenter to orthocenter in a 2:1 ratio.
The circumcenter is at the midpoint of the hypotenuse, and the orthocenter is at the vertex of the right angle. Use this as a practical reminder before finalizing the result.
The inradius r = Area / s, where s is the semi-perimeter (a+b+c)/2. The incenter is the center of this inscribed circle.