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# Centroid

In geometry, a triangle is quite an elementary figure. It is a three-sided bounded figure. A triangle has 3 interior angles. Triangles can be classified on the basis of number of sides as well as number of angles, such as - equilateral, isosceles, scalene and acute-angled, obtuse-angled, right-angled triangle. There are various concepts related to triangles. Centroid is an important property of a triangle. The centroid of a triangle is the center of the triangle. It is defined as the point of intersection of all the three medians of a triangle; where, a median is a line segment joining the midpoint of a side and opposite vertex.

If we have an object, then we can say that the centroid of that object is its center. The centroid of a triangle is the center point of it where its center of gravity is located.
The centroid of the triangle separates the medians in the ratio 2:1 ; i.e. the length of the portion of median from vertex to centroid is twice the length of portion of median from centroid to the midpoint. We come across several concepts and formulae related to centroid of a triangle. It is quite commonly used concept in geometry. In this page, we will learn more about centroid of a triangle and various applications related to it.

 Related Calculators Centroid Calculator

## Centroid Definition

Centroid is a point where all the three medians of the triangle intersect.

The centroid of triangle can be got by finding the average of the x-coordinate's value and the average of the y-coordinate's value of all the vertices of the triangle.

Centroid size: Centroid of any geometric figure is a point. And so, it looks like a dot.

## Centroid Formula

In the figure shown below, the three vertices of the triangle are $A(x_1, y_1)$ , $B(x_2, y_2)$ and $C(x_3, y_3)$.

### Centroid Method

Centroid of points, A, B and C is $\frac{x_1 + x_2 + x_3}{3}$, $\frac{y_1 + y_2 + y_3}{3}$.

## Properties of Centroid

Below are following properties of the centroid:
• Centroid is a term that describes the center of an object.
• It is also called as center of gravity, the geocenter and the barycenter.
• The centroid is always in the interior of the triangle.

## Median Centroid

A median of a triangle is a line segment joining a vertex and the midpoint of the opposite side. The three medians $m_a$, $m_b$, and $m_c$, intersect in a single point, G, the triangle's centroid.

## List of Centroids

Below is centroid formulas for some geometric figures:

### Centroid of Trapezoid

A trapezoid is a quadrilateral with two sides parallel. Let a and b be the parallel sides. The formula used to find the centroid of a trapezoid with parallel sides and the centroid is located a distance of x above b:

x = $\frac{b + 2a}{3(a+b)}$ * h

where, base is b and h is height.

### Centroid of Semicircle

A semicircle is one-half of a full circle.

The centroid of the semicircular = $\frac{4 r}{3\pi}$.

where, r is the radius of the semicircle.

### Centroid of Circle

The intersection of any two diameters, i.e., the centre of the circle is the centroid of the circle.

or centroid of the circle = 2(The centroid of the semicircular) = 2 * $\frac{4 r}{3\pi}$.

The centroid of the circle = $\frac{8 r}{3\pi}$.

where, r is the radius of the semicircle.

### Centroid of Rectangle

Area in the shape of a rectangle have two axes of symmetry and their intersection locates the centroid.

### Centroid of Cone

The centroid of a cone is located on the line segment that connects the apex to the centroid of the base.

Centroid location in cone: Centroid of the cone is $\frac{1}{4}$ the distance from the base to the apex. For a hollow cone, the centroid is $\frac{1}{3}$ the distance from the base plane to the apex.

## Centroid Table

In this table, we can see the centroids of some plane geometric shapes like centroid of parabola, centroid quarter circle, centroid of arc (circle) and centroid of polygon.

### Centroid of Shapes

 Shapes Figure $\bar {x}$ $\bar{y}$ Area 1 Triangle - $\frac{h}{3}$ $\frac{bh}{2}$ 2 Semicircle 0 $\frac{4R}{3\pi}$ $\frac{1}{2}$$\pi R^2 3 Quarter circle \frac{4R}{3\pi} \frac{4R}{3\pi} \frac{1}{4}$$\pi R^2$ 4 Sector of a circle $\frac{2R}{3 a}$ sin a 0 a$R^2$ 5 Parabola 0 $\frac{3h}{5}$ $\frac{4ah}{3}$ 6 Semi Parabola $\frac{3a}{8}$ $\frac{3h}{5}$ $\frac{2ah}{3}$ 7 Arc of Circle Points on the curve, r = $\rho$ from $\theta$ = -$\alpha$ to $\theta$ = $\alpha$ $\frac{\rho sin \alpha}{\alpha}$ 0 2 $\alpha$$\rho$ 8 Right Triangular $\frac{-b}{3}$ $\frac{h}{3}$ $\frac{bh}{2}$ 9 Parabolic spandrel $\frac{3a}{4}$ $\frac{3h}{10}$ $\frac{ah}{3}$

## Equilateral Triangle Centroid

The centroid of a triangle is the point where the triangle's medians intersect. The centroid is always inside the triangle. In the case of an equilateral triangle, incenter, circumcenter, centroid and orthocentre occur at the same point.

## Centroid of Tetrahedron

Tetrahedron is a polyhedron composed of 4 triangular faces, 3 of which meet at each vertex. To find the coordinates of the centroid of the tetrahedron whose vertices are ($x_1, y_1, z_1$ ), ($x_2, y_2, z_2$), ($x_3, y_3, z_3$ ) and ($x_4, y_4, z_4$ ).

Let P($x_1, y_1, z_1$ ), Q($x_2, y_2, z_2$), R($x_3, y_3, z_3$ ) and S($x_4, y_4, z_4$ ) be the vertices of tetrahedron PQRS.

If $G_1$ is the centroid, then its coordinates are (x, y, z).

where, x = $\frac{x_1 + x_2 + x_3 + x_4}{4}$

y = $\frac{y_1 + y_2 + y_3 + y_4}{4}$

z = $\frac{z_1 + z_2 + z_3 + z_4}{4}$

Let us find the coordinates of the centroid of the tetrahedron with vertices (1, 0, 0), (0, 2, 0), (0, 4, 1) and (0, 0, 3).

Coordinates of the centroid = ($\frac{1 + 0 + 0 + 0}{4}, \frac{0 + 2 + 4 + 0}{4}, \frac{0 + 0 + 1 + 3}{4}$)

= $(\frac{1}{4}, \frac{6}{4}, \frac{4}{4})$

= (0.25, 1.5, 1)

## Centroid Construction

The centroid of a triangle is the point where its medians intersect. Let us construct the centroid of a triangle with compass and straightedge.

Step 1: Draw a triangle, ABC, with the help of straightedge.

Step 2: Draw a median for AB, using centres A and B. Mark the midpoint as D and join CD.

Step 3: Draw a median for AC, using centres A and C. Mark the midpoint as F and join BF.

Step 4: Draw a median for BC, using centres B and C. Mark the midpoint as E and join AE.

The intersection of the three medians is the centroid.

## Finding Centroid

Given below are few examples based on centroid of a triangle.

### Solved Examples

Question 1: Calculate the centroid coordinates of the triangle whose vertices are A(4, 8), B(2, 6) and C(0, 10).
Solution:

Given that the vertices are A(4, 8), B(2, 6) and C(0, 10)

A(4, 8) is $A(x_1, y_1)$

C(0, 10) is $C(x_3, y_3)$

The formula to calculate the centroid of the triangle is $(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3})$.

= $(\frac{4 + 2 + 0}{3}, \frac{8 + 6 + 10}{3})$

= $(\frac{6}{3}, \frac{24}{3})$

= (2, 8)

The centroid coordinates of the triangle with vertices A(4, 8), B(2, 6) and C(0, 10) is (2, 8).

Question 2: Calculate the centroid of the triangle whose vertices are A(5, 4), B(6, 1) and C(9, 0).
Solution:

Given that the vertices are A(5, 4), B(6, 1) and C(9, 0).

A(5, 4) is $A(x_1, y_1)$

B(6, 1) is $B(x_2, y_2)$

C(9, 0) is $C(x_3, y_3)$

The formula to calculate the centroid of the triangle is $(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3})$.

= $(\frac{5 + 6 + 9}{3}, \frac{4 + 1 + 0}{3})$

= $(\frac{20}{3}, \frac{5}{3})$

= (6.67, 1.67)

The centroid coordinates of the triangle with vertices A(5, 4), B(6, 1) and C(9, 0)is (6.67, 1.67).