# Centroid

The central point of the item is referred to as the centroid of that object. The centroid is the center of the object's average location. If we cut a piece from any object, we can precisely balance the cut-out shape on its centroid. Centroid can also be defined as a point in any 2D figure where the total displacement of all the points converges to zero.

## Properties of Centroids

Listed below are some basic yet important properties of a centroid in any shape. They are:

• A centroid always lies inside the object; it can never be outside the shape.
• The center of gravity lies at the centroid; this means that you can balance anything at the centroid.
• If we draw all the medians respective to the object, they will always converge at the centroid.
• In easy-to-understand language, a centroid is the center of any object.

## List of Popular Shapes with their Centroid

### The Centroid of a Triangle:

The center of the triangle is known as the point of intersection of the three medians. The intersection between all the three media is also specified. The median is a line connecting the center point of the triangle with the opposing vertex. Each median is divided with a ratio of 2:1 by the centroid. This is obtained by finding the average x- and y-coordinate points of all the vertices of a triangle.

Let us consider a triangle ABC with medians from all the three vertices to their respective opposite sides. Let the median from A be AK, from B be BL, and from C be CM (refer to the figure). The point where these medians meet mark that as O; the point O is known as the centroid of the triangle.

Formula: Let the triangle ABC be in a cartesian plane with coordinates of A = (a1, b1), coordinates of B = (a2, b2) and coordinates of point C (a3, b3). The centroid can be determined by finding the average of respective coordinates. Therefore, coordinates of point O will be X = (a1 + a2 + a3)/3 and Y = (b1 + b2 + b3)/3.

### The Centroid of a square

The point of interaction of the two diagonals of a square is the center point or centroid of the square. For example, let us consider a square PQRS with PR and QS as the diagonals of the square. The point O where these two diagonals meet is the point of the centroid. Most of the quadrilaterals possess the same concept for their centroid.

### Orthocenters and Incenters

We must not confuse centroid with orthocenters and incenters. An orthocenter is a point of intersection of the altitudes or height of any object. The orthocenter may even be present outside the object. For instance, the orthocentre of the obtuse triangle and parallelogram lies outside the shape. The incenter, on the other hand, is the point where the angle bisectors of shapes like triangle and quadrilateral meet. This point always lies inside the object, and there is no fixed ratio of how the incenters and orthocenters will divide each other.

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