Math Calculators ▶ Centroid Calculator

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An online centroid calculator helps you to find the centroid of a triangle (ABC), N-Points, and N-sided polygon for the given coordinates. No doubt, this calculator will best for finding the centroid of many 2D shapes, as well as of a set of points.

**NOTE THAT!**

This centroid of a triangle calculator can find the centroid of a triangle, rectangle, trapezoid, kite, or any other imaginable shape, but the only restriction is: you should need to make sure that “the polygon should be closed, consists of maximum ten vertices, and non-self-intersecting.”

Now, you are going to get here a basic centroid definition, centroid formula, how to find centroid (step-by-step) and with a calculator, and different centroid-related terms. Let’s find the terms!

The point through which all the three medians of a triangle pass is said to be the centroid of a triangle. More specifically, if you draw lines from each corner (or vertex) of a triangle t the midpoint of the opposite sides, then those three lines meet at a center or centroid of the triangle. It is also said to be ‘center of gravity of triangle’ where the triangle balances evenly, ‘center of mass of a triangle, or barycenter.’ Keep in mind; centroid has several numbers of properties and relations with other parts of the triangle that includes its circumcenter, incenter, orthocenter, area, and much more! Simply, try centroid calculator to determine the coordinates on the centroid of a triangle. Also, use this online midpoint calculator (finder) that helps to find the midpoint and distance of a line segment and also provides you with the step-by-step calculations.

find the distance and midpoint of a line segment and shows you the step-by-step calculations. T

Simply, the coordinates of the centroid are the average of the coordinates of the vertices. So, if you want to find the x coordinates of the orthocenter, then you ought to add up the three vertex x coordinates and divide by 3: repeat, the same for the y coordinate.

**The centroid of a triangle formula is:**

According to centroid theorem, the centroid equation is stated as:

{(x1+x2+x3)/3, (y1+y2+y3)/3}

For special triangles, you can find the centroid quite readily:

If the side length is known, you can readily find the centroid of an equilateral triangle:

G = (a/2, a√3/6)

If an isosceles triangle has legs of length ‘l’ and height ‘h’, then the centroid is:

G = (l/2, h/3)

For a right triangle, if the two legs ‘b’ and ‘h’ are given, then you can readily find the right centroid formula straight away!

G = (b/3, h/3)

The first thing that you have to remember that centroid is the center point equidistant from all vertices.

Now let’s start with the formula:

Ox = x1+x2+x3/3

Oy = y1+y2+y3/3

**For example:**

Suppose that abc triangle has vertices A = (4,5), B = (20,25), and C = (30,6). How to find coordinates of the centroid?

**Solution:**

- First of all, you have to identify the coordinates of each vertex in the triangle, in the above example, the vertices are A = (4,5), B = (20,25), and C = (30,6).
- Very next, you have to add all the x values from the three vertices coordinates and divide by 3 to get the x value of the centroid coordinate.

Ox = 4 + 20 + 30 / 3

Ox = 54 / 3

Ox = 18

- Then, you have to add all the y values from the three vertices coordinates and divide by 3 to get the y value of the centroid coordinate.

Oy = 5 + 25 + 6 / 3

Oy = 36 / 3

Oy = 12

- Finally, you determined the centroid coordinate from the above calculations, centroid coordinate = (18, 12)

So, you can draw this centroid into the triangle to illustrate its center position. However, if you don’t want to do it manually, then use our centroid calculator that helps to find the centorid of a trinagle or different 2D shapes.

This centroid of triangle calculator enables you to find the coordinates on the centroid of a triangle ABC, the centroid of any N-Points, and N-sided Polygon. When the vertices are known, the centroid calculator will easily find centroid of any 2-D shape.

Sometimes, this tool is referring as a center of mass calculator, geometric center, or barycenter calculator. More specifically, you can readily find centroid of a triangle or a set of points.

The centroid calculator is specifically designed with user-friendly that determines the centroid of a triangle or any triangle (2D-shapes), when the vertices are given. Let’s take a look!

**Inputs:**

- First of all, you have to select an option of a triangle from the designated box
- Now, all you need to enter the coordinates of the given triangle into the designated fields
- Hit calculate

**Output:**

The centroid calculator shows:

- Centroid of a triangle
- The graph shows the centroid of a triangle

**Inputs:**

- Simply, choose the option of N-sided polygon from the given drop-down
- Now, enter the value for N
- Then, add the values of coordinates into the given fields
- Hit calculate

**Outputs:**

The centroid finder finds:

- Centroid N-sided Polygon
- The graph shows the centroid of N-sided Polygon

**Inputs:**

- First, you ought to choose the N-points option from the given drop down of this calculator
- Then, you ought to add the value for N
- Very next, add the values of coordinates into the designated boxes
- Hit calculate

**Output:**

The calculator find

- Centroid N-Points
- The graph shows the centroid of N-Points

If you want to calculate a polygon’s centroid, G(Cx,Cy) that is defined by its n vertices (x0,y0),(x1,y1),…(xn-1,yn-1), then you just need to use these given formulas:

Fromula:

Formula:

Where:

- A is referred to as a polygon’s signed area:

**Formula:**

Make sure that the vertices should be entered in order and even the polygon should be closed – it indicates that the vertex (x0,y0) is the same as the vertex (xn,yn).

If that centroid formula is hard to remember for you, then use our centroid calculator online!

However, there are certain formulas that help you to find a centroid of a trapezoid on the Internet, the above-mentioned equation is standardized – there’s no need the origin coinciding with one vertex, nor the trapezoid base in line along with the x-axis.

All you need to know the vertices to find the centroid position. The same equation is applied to the following:

- centroid of a rectangle
- centroid of a rhombus
- centroid of a parallelogram
- centroid of a pentagon
- centroid of any other closed, non-self-intersecting polygon

If you need to find the centroid of N points, then you ought to calculate the average of their coordinates:

Gx = (x1 + x2 + x3 +… + xN) / N

Don’t fret; this is the same general formula that we already mentioned-above!

Detertming the centroid of a simple set of points is taken into account in several real-life applications like date analysis. The well-known method is said to be the K-means clustering, in which an algorithm tries to minimize the squared distance between the data points and also the cluster’s centroids.

Let’s make a comparison between centroid and center of gravity:

**Centroid:**

- Centroid is said to be the geometric center of the object
- Centroid is indicated with the use of the letter ‘c’
- Centroid can readily be calculated by using the plumb line method or simply by taking the mean of the media, specifically for a triangle
- Centroid is referred to as the central point of objects with uniform density
- The term of centroid is often used in Mathematics, in relation to triangles
- Generally, Centroid deals with 2D structures

**Centre of Gravity:**

- Centre of Gravity is said to be the point where the total weight of the object acts
- Centre of Gravity is represented as the use of the letter ‘g’
- Centre of Gravity can readily be calculated by using this equation W=S x dw
- Centre of Gravity can be applicable to objects with any density
- The term of Centre of Gravity is typically used in Physics
- Generally, Centre of Gravity deals with 3d structure

Let’s, now look at the example for better understanding!

**Example:**

Suppose that you have cardboard of length ‘l’ and breadth ‘b’, both the centre of gravity and centroid will rely on the same point where the two diagonals meet. Now, you just ought to make the hole in any side of the cardboard and just fit the steel of the same dimension into that hole. So, here the centroid of the body will remain the same as it was previously, but the position of centre of gravity changes.

Read on!

Remember that centroids offer balancing points for triangles, so they are crucial points for artists who want to build mobiles, or moving sculptures. You can easily make such a mobile yourself by simply using wire, string, or fishing line, and even various sizes of triangles that cut from stiff plastic, cardboard, or thin wood.

You just ought to paint each triangle a bright color, then simply tie each triangle by its centroid to a wire. Here, the wire can be suspended from another wire, and so on, until you attain a balanced mobile. Remember that each triangle will glide through the air completely flat as the centroid is its balancing point.

Aircraft should be perfectly balanced around their center of gravity (CG), or centroid for the pilot to maintain control. There are several factors that influence the pilot’s ability to control the airplane’s motion by using three different axes, but if the airplane is not engineered accurately to balance around its centroid or CG, then no amount of pilot control will be enough to keep the plane flying correctly.

The centre of gravity (CG) of an airplane applies whether you’re building a radio-controlled plane, a model aircraft, or an actual military or passenger jet.

The centroid of an equilateral triangle can readily find as it is always located inside the triangle like the (incenter, another one the triangle’s concurrent points). Remember that the centroid divides each median in a ratio of 2:1. More specifically, the centroid will always be 2/3 of the way with any given median towards the vertex, and 1/3 towards the side.

The point where all three altitudes of the triangle intersect is said to be as the orthocenter of a triangle. A line that passes through a vertex of the triangle and perpendicular to the opposite side is known as altitude. Thus, there are three altitudes in a triangle.

- The intersection of the medians always forms centroid
- It is said to be as one of the points of concurrency of a triangle
- It is always located inside the triangle
- Centroid divides each median in a ratio of 2:1. In other words, it will always be 2/3 of the way along any given median

By definition, the centroid is said to be a point of a concurrency of the triangle. It represents the point where all 3 medians intersect and are typically described as the barycent or the triangle’s center of gravity. It always formed by the intersection of the medians. You can use this mathematical centroid calculator to find the point of a concurrency of the triangle.

In the terms of mathematics and physics, the centroid or geometric center of a plane figure is referred to as the arithmetic mean position of all the points in the figure. Informally, it is said to be as the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.

Remember that if a shape possesses an axis of symmetry, then its centroid point will always be located on that axis. Moreover, it is possible for the centroid of an object to be located outside of its geometric boundaries.

Centroids are said to be most useful for studying centers of gravity and even the moments of inertia in physics and engineering. So, logically, it seems that the centroid should remain within the triangle, and only the irregular shapes with extended sides consist of centers of gravity on the exterior.

When it comes to finding the centroid of a 3D shape, you have to looking for x, y, and z coordinates (x̄, ȳ, and z̄). This be referred to as the x, y, and z coordinate of the point, which is the centroid of the shape.

The centroid is also referred to as the center of gravity of the triangle. Well, if you have a triangle plate, then simply try to balance the plate on your finger. Once you got the point where it will balance, that is said to be the centroid of that triangle.

When it comes to nouns, centre indicates the point in the interior of a circle, which is equidistant from all points on the circumference, while centroid referred to as the point at the centre of any shape, sometimes said to be as the centre of area or centre of volume. When it comes to a triangle, the centroid is referred to as the point at which the medians intersect.

Get ease with the above centroid of a triangle calculator, formula, and solved example to understand, practice, and verify such geometrical calculations. Unlike most of the other online calculators, it provides you with the tested mathematical calculations for finding the centroid of a trinagle, centroid of N-Sided Polygons, and N-Points.

From the source of Wikipedia: Centroid, Properties, Locating (Plumb line method) and much more

From the source of askanydifference: Centre of Gravity vs Centroid (In Tabular Form)

From the source of artofproblemsolving: Proof of concurrency of the medians of a triangle