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Centroid Calculator

Select your desired geometrical figure and enter its coordinates. The calculator will determine the centroid and display detailed calculations.

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Coordinates (x₁, y₁)

Coordinates (x₂, y₂)

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Centroid Calculator

Calculate the centroid of a triangle, other 2D figures, or a set of points. The centroid calculator computes the point where the medians intersect in just a second. You can also visualize the shape and the centroid for better understanding. The calculator supports polygons with up to 10 vertices that are non-intersecting and closed.

What Is the Centroid of a Triangle?

"The point through which all three medians of a triangle pass is called the centroid of the triangle."

This concept is closely related to the midpoint of a line segment.

General Formula for the Centroid of a Triangle

For a triangle with vertices:

\(A = (x_1, y_1), B = (x_2, y_2), C = (x_3, y_3)\)

The centroid \(G\) is calculated as the average of the x-coordinates and y-coordinates:

\(G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\)

Centroid Formulas for Special Triangles

Equilateral Triangle: If the side length is \(a\), then
\(G = \left( \frac{a}{2}, \frac{a \sqrt{3}}{6} \right)\)
Isosceles Triangle: If the legs are length \(l\) and height \(h\), then
\(G = \left( \frac{l}{2}, \frac{h}{3} \right)\)
Right Triangle: With legs \(b\) and \(h\),
\(G = \left( \frac{b}{3}, \frac{h}{3} \right)\)

Centroid of a Polygon

For a closed polygon (vertex \((x_0, y_0) = (x_n, y_n)\)), the centroid is given by:

\(C_x = \frac{1}{6A} \sum_{i=0}^{n-1} (x_i + x_{i+1}) (x_i y_{i+1} - x_{i+1} y_i)\)

\(C_y = \frac{1}{6A} \sum_{i=0}^{n-1} (y_i + y_{i+1}) (x_i y_{i+1} - x_{i+1} y_i)\)

where the area \(A\) of the polygon is:

\(A = \frac{1}{2} \sum_{i=0}^{n-1} (x_i y_{i+1} - x_{i+1} y_i)\)

Centroid of a Set of Points

For \(N\) points, the centroid is the average of all x- and y-coordinates:

\(G_x = \frac{x_1 + x_2 + \dots + x_N}{N}, \quad G_y = \frac{y_1 + y_2 + \dots + y_N}{N}\)

Example: Centroid of a Triangle

Consider triangle ABC with vertices:

A = (4,5)
B = (20,25)
C = (30,6)

Centroid calculation:

\(G = \left( \frac{4+20+30}{3}, \frac{5+25+6}{3} \right) = \left( \frac{54}{3}, \frac{36}{3} \right) = (18, 12)\)

Properties of the Centroid

The centroid is the intersection of the medians of a triangle.
It is one of the points of concurrency of a triangle.
The centroid always lies inside the triangle.
It divides each median in a ratio of 2:1 (i.e., 2/3 of the distance from a vertex along the median).

FAQs

Can a centroid be outside of a shape?

Yes. For shapes with an axis of symmetry, the centroid lies on that axis. For certain concave or irregular shapes, the centroid can be outside the geometric boundaries.

How do you find the centroid of a 3D object?

For a 3D object, you need to find the average of x, y, and z coordinates: \( (\bar{x}, \bar{y}, \bar{z}) \). This point is the centroid of the 3D object.

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