How do you find the centroid of a triangle?

The centroid G is the average of the three vertex coordinates: G = ((x1+x2+x3)/3, (y1+y2+y3)/3). It is the intersection point of the three medians and divides each median in a 2:1 ratio.

What is the distance from centroid to vertex?

The distance from centroid G to each vertex is two-thirds of the median length from that vertex. Use the distance formula: d = sqrt((xi-xG)^2 + (yi-yG)^2). The three distances are generally different unless the triangle is equilateral.

Why does the centroid divide medians in a 2:1 ratio?

The centroid is the balance point of the triangle. It splits each median so that the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side. This is a fundamental property of medians.

Are the centroid distances equal for all triangles?

Only in equilateral triangles are all three centroid-to-vertex distances equal. In scalene and isosceles triangles, the distances vary. The centroid is always closest to the vertex opposite the longest side.

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Home›Geometry Tools›Centroid to Vertex Distance Calculator

Centroid to Vertex Distance Calculator

Enter three vertices to compute centroid coordinates and distances to each vertex

Vertex A x1

Vertex A y1

Vertex B x2

Vertex B y2

Vertex C x3

Vertex C y3

Centroid Distance Formula

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

Distance AG = sqrt((x1-xG)^2 + (y1-yG)^2)

Distance BG = sqrt((x2-xG)^2 + (y2-yG)^2)

Distance CG = sqrt((x3-xG)^2 + (y3-yG)^2)

The centroid is the geometric center of a triangle, also known as the center of mass. Each centroid-to-vertex distance equals two-thirds of the corresponding median length.

⚠ Vertices should form a non-degenerate triangle (not collinear). The centroid always lies inside the triangle.

What Is the Centroid?

The centroid of a triangle is the intersection point of its three medians. It serves as the triangle's center of mass and balance point. It divides each median in a 2:1 ratio (vertex to centroid : centroid to midpoint).

Coordinates

G = (average of x, average of y). Simplest of all triangle centers to compute.

2:1 Ratio

The centroid splits each median so that the vertex segment is twice as long as the midpoint segment.

Distance Formula

Use standard Euclidean distance: d = sqrt(dx^2 + dy^2). Always produces positive distances.

Equilateral Case

In an equilateral triangle, all three centroid distances are equal: the circumradius.

Teaching Example: Triangle A(0,0), B(6,0), C(3,4). Centroid G = ((0+6+3)/3, (0+0+4)/3) = (3, 1.333). AG = sqrt((0-3)^2+(0-1.333)^2) = 3.33. BG = sqrt((6-3)^2+(0-1.333)^2) = 3.33. CG = sqrt((3-3)^2+(4-1.333)^2) = 2.67.

Applications

Physics Center of Mass Structural Analysis Computer Graphics Robotics Navigation Engineering Design

FAQs about Centroid Distances

How to find the centroid of a triangle?▼

Average the x-coordinates and y-coordinates of the three vertices: G = ((x1+x2+x3)/3, (y1+y2+y3)/3).

Does the centroid always lie inside the triangle?▼

Yes, the centroid is always inside the triangle for all triangle types (acute, right, obtuse). It is the only triangle center guaranteed to be inside.

Are the three distances always equal?▼

Only in equilateral triangles are all three distances equal. In other triangles, the distances vary by vertex. The centroid is closest to the vertex opposite the longest side.

What is the 2:1 centroid rule?▼

The centroid divides each median in a 2:1 ratio. The distance from vertex to centroid is twice the distance from centroid to the midpoint of the opposite side.

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