Centroid Calculator

The centroid calculator finds the geometric center of a triangle, polygon, or any set of points by averaging their coordinates. You enter the x and y values for each vertex, and the tool returns the centroid coordinates Gx and Gy immediately. It works for triangles (3 points), custom polygons (N sides), and even freeform N-point sets — making it useful for geometry, physics, and engineering problems alike.

What Is Centroid?

The centroid is the arithmetic mean position of all points in a plane figure. In simpler terms, it is the geometric center of a shape — the single point that represents the average location of every point within the figure.

For a triangle specifically, the centroid is the point where all three medians intersect. A median is a line drawn from a vertex to the midpoint of the opposite side. Every triangle has exactly three medians, and they always meet at one point — the centroid.

Here are the key properties of a centroid:

• It is always located inside the triangle, no matter if the triangle is acute, obtuse, or right-angled.

• It divides each median in a 2:1 ratio — the longer segment is always on the vertex side.

• For a uniform 2D shape, it acts as the exact balance point or center of mass.

• It is a purely geometric concept — unlike center of gravity, it does not depend on mass or weight distribution.

In engineering and physics, centroids are used to calculate moments of inertia, structural load distribution, and centers of gravity for uniform objects.

The Centroid Formula

For a triangle with three vertices, the centroid (G) is calculated by averaging the x-coordinates and the y-coordinates of all three points separately.

Formula for a Triangle

Gx = (x₁ + x₂ + x₃) ÷ 3

Gy = (y₁ + y₂ + y₃) ÷ 3

Where (x₁, y₁), (x₂, y₂), and (x₃, y₃) are the coordinates of the three vertices.

Formula for an N-Sided Polygon or N-Point Set

Gx = (x₁ + x₂ + ... + xₙ) ÷ n

Gy = (y₁ + y₂ + ... + yₙ) ÷ n

The centroid is simply the average of all x-coordinates and the average of all y-coordinates across all n points.

How the Centroid Calculator Works

The calculator lets you define a shape first, then enter coordinates for each point.

Inputs

• Define a shape — A dropdown lets you choose between Triangle, N-sided polygon, or N-point set. The default is Triangle.

• Point 1, Point 2, Point 3 — Two fields per point for the x-coordinate (left field) and y-coordinate (right field). Default values are Point 1 (0, 0), Point 2 (0, 3), Point 3 (3, 3).

Outputs

• Centroid (G) — Two output values: Gx (horizontal center) and Gy (vertical center), displayed in blue.

• A note below the output reads: "Computed as average of 3 triangle vertices."

The calculator handles the averaging automatically. For polygon or N-point mode, additional point fields appear as needed.

How to Calculate Centroid — Step by Step

Example 1: Using the Default Calculator Values

The calculator's default inputs are Point 1 (0, 0), Point 2 (0, 3), and Point 3 (3, 3).

Gx = (0 + 0 + 3) ÷ 3 = 3 ÷ 3 = 1

Gy = (0 + 3 + 3) ÷ 3 = 6 ÷ 3 = 2

Centroid G = (1, 2) — which matches the calculator output exactly.

Example 2: A Different Triangle

Suppose your triangle has vertices at Point 1 (2, 4), Point 2 (6, 2), and Point 3 (4, 8).

Gx = (2 + 6 + 4) ÷ 3 = 12 ÷ 3 = 4

Gy = (4 + 2 + 8) ÷ 3 = 14 ÷ 3 = 4.67

Centroid G = (4, 4.67). Enter these coordinates into the centroid calculation tool and you'll get the same result instantly.

Centroid Reference Table

Here are the centroids for some common triangles to use as a quick reference:

Note: All Gx and Gy values calculated using (sum of coordinates) ÷ 3. Results rounded to two decimal places where applicable.

Where Centroid Calculations Actually Matter

Centroid calculation shows up in more fields than just geometry class.

Structural engineering — Engineers find the centroid of a beam's cross-section to determine where bending stresses are distributed. Getting this wrong leads to structural miscalculations.

Computer graphics and image processing — The centroid of a detected shape or blob helps software identify object positions in an image. This is used in robotics, facial recognition, and motion tracking.

Physics — For uniform objects, the centroid equals the center of mass. This matters when calculating torque, rotational motion, and equilibrium.

Architecture and design — Architects use centroids to balance visual weight and find geometric anchors in floor plans or structural layouts.

Fun Fact

The centroid of a triangle is sometimes called the "median point" — but here's what most people don't know: if you cut a triangle out of cardboard and balance it on a pencil tip, the only point it will balance perfectly is the centroid. Not the center of the longest side. Not the midpoint of the triangle's height. Exactly at Gx and Gy. Ancient builders intuitively used this property to balance stone slabs before the formula was ever written down. 😄

Frequently Asked Questions

What is a centroid?

The centroid is the geometric center of a shape, calculated as the average position of all its points. For a triangle, it is the point where the three medians intersect. It acts as the balance point for any uniform flat shape.

How do you calculate the centroid of a triangle?

Add the x-coordinates of all three vertices and divide by 3 to get Gx. Do the same for the y-coordinates to get Gy. The result (Gx, Gy) is the centroid. For example, vertices (0,0), (0,3), (3,3) give a centroid of (1, 2).

What is the 1/3 2/3 rule for the centroid?

The centroid divides each median of a triangle in a 2:1 ratio. This means the centroid sits two-thirds of the distance from any vertex toward the midpoint of the opposite side, and one-third of the distance from that midpoint back to the vertex. This is often called the 1/3–2/3 rule.

What is the centroid theorem in class 9 geometry?

The centroid theorem states that the centroid of a triangle divides each median in a 2:1 ratio, with the longer part always being on the side of the vertex. It also establishes that all three medians of any triangle always intersect at exactly one point — the centroid.

Can the centroid be outside the triangle?

No. The centroid of a triangle is always located inside the triangle, regardless of its type — acute, obtuse, or right-angled. This makes it different from other triangle centers like the orthocenter or circumcenter, which can fall outside the triangle.