Centroid of a Triangle Calculator

Find the centroid coordinates, median lengths, and area of a triangle from its three vertices.

Enter Triangle Vertices

Vertex A

Vertex B

Vertex C

Result

Centroid (G)
(3, 3)
coordinate units
Centroid x (Gx) --
Centroid y (Gy) --
Triangle Area --
Perimeter --
Median from A --
Median from B --
Median from C --

Step-by-Step Solution

G = ((x1+x2+x3)/3, (y1+y2+y3)/3)

What Is the Centroid of a Triangle?

The centroid of a triangle is the point where all three medians of the triangle intersect. A median is a line segment joining a vertex to the midpoint of the opposite side. The centroid is often denoted by G and is sometimes called the "center of gravity" of the triangle because a uniform triangular lamina would balance at this point.

The centroid always lies inside the triangle, regardless of the type of triangle (acute, right, or obtuse). This makes it distinct from other triangle centers like the circumcenter (which can lie outside for obtuse triangles) or the orthocenter.

Centroid Formulas and Properties

Centroid Coordinates

The centroid is the arithmetic mean of the vertex coordinates.

G = ((x1+x2+x3)/3, (y1+y2+y3)/3)

Median Length

Length of a median from vertex to midpoint of opposite side.

m_a = sqrt(2b^2 + 2c^2 - a^2) / 2

2:1 Division Property

The centroid divides each median in a 2:1 ratio from the vertex.

AG = (2/3) * m_a

Triangle Area

Area from coordinates using the cross-product formula.

A = |x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|/2

How to Find the Centroid of a Triangle

  1. Identify the vertices: Record the three vertices A(x1, y1), B(x2, y2), and C(x3, y3).
  2. Compute the centroid x-coordinate: Gx = (x1 + x2 + x3) / 3.
  3. Compute the centroid y-coordinate: Gy = (y1 + y2 + y3) / 3.
  4. Find the midpoints: The midpoint of side BC is M_a = ((x2+x3)/2, (y2+y3)/2), and similarly for the other sides.
  5. Calculate median lengths: The median from vertex A to midpoint M_a has length equal to the distance from A to M_a.

The Median and the 2:1 Ratio

Each median of a triangle connects a vertex to the midpoint of the opposite side. The centroid divides each median into two segments: the segment from the vertex to the centroid is twice as long as the segment from the centroid to the midpoint. This 2:1 ratio is a fundamental property of the centroid and can be used to locate the centroid once the medians are drawn.

Other Triangle Centers

The centroid is just one of several important triangle centers:

  • Incenter: The point where angle bisectors meet, equidistant from all three sides. Center of the inscribed circle.
  • Circumcenter: The point where perpendicular bisectors of the sides meet, equidistant from all three vertices. Center of the circumscribed circle.
  • Orthocenter: The point where the altitudes (heights) of the triangle intersect.
  • Euler line: The centroid, circumcenter, and orthocenter are all collinear (they lie on the same line), known as the Euler line. The centroid is located 1/3 of the way from the circumcenter to the orthocenter.

Real-World Applications

  • Architecture: Finding the balance point of triangular structural elements like trusses and gables.
  • Physics: Determining the center of mass of a uniform triangular plate.
  • Computer Graphics: Calculating face normals and texture coordinates for triangular meshes.
  • Navigation: Finding the centroid of a triangular region on a map for positioning labels or markers.