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Centroid of a Triangle Calculator

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This centroid of a triangle calculator will return the location of the centroid for your triangle. A centroid is a point where the center of gravity lies for any object with uniform mass distribution. The concept of the center of mass is well explored in the areas of mechanics, particularly in the strength of materials.

In this article, we will explore the centroid of a triangle. Read on to understand what is a centroid and how to find the centroid of a triangle.

What is a centroid?

In a triangle, if you take the average or arithmetic mean of coordinates of all the points, you'll get the center point of the geometry. This center point is known as the centroid. Mathematically, for a triangle with points (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), and (x3,y3)(x_3, y_3). The ordinate, ycy_c of the centroid for the polygon is:

yc=y1+y2+y33=13∑n=13yiy_c = \frac{y_1 + y_2 + y_3}{3} = \frac{1}{3}\sum_{n=1}^3 y_i

Similarly, the abscissa or the x coordinate for the centroid is

xc=x1+x2+x33=13∑n=13xix_c = \frac{x_1 + x_2 + x_3}{3} = \frac{1}{3}\sum_{n=1}^3 x_i

The equations for the centroid, C having coordinates (xc,yc)(x_c, y_c) are the centroid formulae for a triangle.

Alternatively, you can also find the centroid of a triangle by drawing medians. Geometrically, a centroid is a point that lies at the intersection of medians of the triangle. Such that a centroid of a right triangle is one-thirds of its height and base, i.e.,

xc=b3;yc=h3x_c = \frac{b}{3} ; y_c = \frac{h}{3}

How to use centroid of a triangle calculator

Let's find the centroid of a triangle with vertices lying on (1,1)(1,1), (3,4)(3,4), and (4,5)(4,5).

To find the centroid of a triangle with vertices:

  1. Enter the coordinates of point A, x1=1, y1=1x_1 = 1, \ y_1 = 1.
  2. Fill in the coordinates of point B, x2=3, y2=4x_2 = 3, \ y_2 = 4.
  3. Insert the coordinates of point C, x3=4, y3=5x_3 = 4, \ y_3 = 5.
  4. The coordinates are given by the centroid of a triangle calculator as:
xc=1+3+43=2.67yc=1+4+53=3.33\scriptsize \begin{align*} \qquad x_c &= \frac{1 + 3 + 4}{3} = 2.67 \\ y_c &= \frac{1 + 4 + 5}{3} = 3.33 \end{align*}
FAQs

How do I calculate centroid of a triangle?

To calculate the centroid of a triangle:

  1. Add the x-coordinates of all the points.
  2. Divide the sum by 3 to obtain the x-coordinate of the centroid.
  3. Add the y-coordinates of all the points.
  4. Divide the sum by 3 to obtain the y-coordinate of the centroid.

How far is a centroid located from the opposite vertex?

A centroid divides the median into a ratio of 2:12:1, which is why we can also estimate the centroid by traveling one-third of the distance on each side from the opposite vertex.

Triangle in the coordinate system, with vertices and centroid marked.

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