 # How to Calculate the Centroid

Post Contents

In this post we will explore the centroid, it will be full of information in text, equations and pictorial forms with examples that are solved step by step to help you understand and apply equations to calculate the centroid of a shape and the first moment of area and most importantly, why we need too!

## What is the Centroid?

To put it very simply, the centroid is the centre of a shape, such as in a 2×2 square, the centroid of the co-ordinates would be (1, 1). For a shape such as a square it is very easy to find the centroid with simple mathematics, or just through looking at it. However, when we have composite shapes, (two shapes together), or even just more complex shapes in general, the easiest, fastest and most efficient way to calculate the centroid is using an equation.

Being able to calculate the centroid is extremely important for the structural analysis of members, it is involved in the various calculations for different section properties, thankfully, it is really easy to calculate!

## How Do We Calculate the Centroid?

The formula for the centroid is given below, don’t worry if it looks overly complicated, following a breakdown of the variables will we go through a very basic example and it will all make sense. Then we will look at more complex composite shape, after which you will be finding centroids of shapes in your sleep!

## Centroid Formula

X̄*A = ∑(Xi*Ai)

or

ȳ*A = ∑(Yi*Ai)

Here is the breakdown of the variables in the equation for the X-Axis centroid,
X̄ = The location of the centroid in the X Axis
A = The total area of all the shapes
Xi = The distance from the datum or reference axis to the centre of the shape i
Ai = The area of shape i.

## Lets Find the Centroid – Ex. 1

Below is a rectangle, our aim, to find the co-ordinates of the centroid. To help follow through the example the datum or reference axis (Xo & Yo) is put onto the drawing and therefore we should have the same Xi and Yi values.

First thing to note in the example is there is only one shape, we will call this shape 1. To solve the centroid we look at each axis separately, the answers to each provide the co-ordinates (Xi, Yi). Units are not relevant for the centroid.

## X-Axis

X̄*A = X1*A1

So we break down the variables and calculate them step by step.

X̄ = Coordinate Location (Our Answer!)
A  = 5 * 2 = 10
X1 = 2 / 2 = 1
A1 = 5 * 2 = 10

Putting it altogether..

X̄ * 10 = 1 * 10
X̄ = 1

## Y-Axis

ȳ * A = y1 * A1

ȳ = Coordinate Location (Our Answer!)
A  = 5 * 2 = 10
y1 = 5 / 2 = 2.5
A1 = 5 * 2 = 10

Putting it all together..

ȳ * 10 = 2.5 * 10
ȳ = 2.5

Below is the solution, shown graphically with the co-ordinates (X̄  ,ȳ)   of the centroid of the 5×2 rectangle. Simple right? Now lets try a composite shape, which is slightly more complicated.

## Complex Centroid – Ex.2

Below is a composite shape made up of a square and a rectangle, our aim, to find the co-ordinates of the centroid. Once again to help you follow through the example, the datum or reference axis (Xo & Yo) is put onto the drawing and therefore we should have the same Xi and Yi values.

With composite shapes, we need to split the shape into individual shapes (sub-shapes, if you like). From this we can then apply the formulas as above to calculate X-bar and Y-bar for the co-ordinates of the centroid. We need to ensure that the distance from the datum to the centre of the shape runs through all of the composite shape. In this example, we need to split the shape in two different ways. It may sound confusing but with a few pictures it’ll be clear as rain.

## X-Axis

Lets tackle the X axis first. To split it into sub-shapes and ensure that the Xi line follows through both, the solution is given below.

To make it clearer which to solve for, using the equations, below is the shapes separated. As you become more comfortable, you can do this mentally.

Now using the original equation, we can split Xi and Ai according to both shapes, this lets us calculate the area of each shape (A1 and A2). Remember A in the first part of the equation for the whole composite shape!

To calculate X1 and X2 we have to look at the whole shape, as this is the distance between the centroid of A1 or A2 and the datum that we set, X0. See below.

X̄*A = ∑(Xi*Ai)

or

X̄*A = (X1 * A2) + (X2 * A2)

X̄ =  Coordinate Location (Our Answer!)
A  = (1 * 1) + (3 * 1) = 4
X1 = 1 / 2 = 0.5
A1 = 1
X2 = 1+ (1/ 2) = 1.5
A2 = 1 * 3 = 3

X̄ * 4 = 0.5 * 1  + 1.5 * 3
X̄ = 1.25

## Y-Axis

Now we tackle the Y-Axis, to do this we need to split the shape up into different sub-shapes to have a continuous axis running through the whole shape.

Now repeating the same method as completed for the X-axis, we can break the shapes apart to calculate the area.

Before going ahead, see if you can calculate Y1 and Y2 for both shapes.

ȳ =  Coordinate Location (Our Answer!)
A  = (1 * 2) + (2 * 1) = 4
Y1 = 1 / 2 = 0.5
A1 = 1 * 2 = 2
Y2 = 1+ (2 / 2) = 2
A2 = 1 * 2 = 2

ȳ * 4 = (0.5 * 2) + (2 * 2)
ȳ = 1.25

Putting both X-bar and Y-bar together, we get the co-ordinates of (1.25, 1.25) for the centroid of the composite shape.

## Video Example

So, this concludes the end of the tutorial on how to calculate the centroid of any shape. You will find out how useful and powerful knowing how to calculate the centroid can be, in particular when assessing the shear capacity of an object using the first moment of area.

Subscribe to our mailing list
* indicates required

1. Ƭhis page cеrtainly has all of the info I wanted
concerning this subject and didn’t know who to ask.

2. Hi friends, how is the whole thing, and what you desire to
say on the topic of this post, in my view its genuinely remarkable
designed for me.

3. I’m pretty pleased to uncover this web site. I need to to thank
I definitely loved every part of it and I have you saved as a favorite to see
new information in your web site.

4. Hi there, You have done a great job. I’ll definitely digg it and
personally recommend to my friends. I’m confident they will be benefited from this website.

5. that you simply shared this useful info with us. Please keep us informed like this.
Thank you for sharing.

6. Quality posts is the important to attract the people to visit
the site, that’s what this website is providing.

7. Simply wish to say your article is as astounding.
The clearness to your put up is simply excellent and i could
think you are an expert in this subject. Fine with your
permission allow me to snatch your RSS feed to keep updated with coming near near
post. Thank you a million and please carry on the gratifying
work.

8. I’m impressed, I must say. Rarely do I encounter a blog that’s both equally educative
and entertaining, and let me tell you, you’ve hit
the nail on the head. The problem is an issue that too few
people are speaking intelligently about. Now i’m very happy
I found this in my search for something relating to this.

9. I wish my site loaded up as quickly as yours lol

1. Author

Hi Solomon,
Tom

10. I enjoy, result in I found exactly what I used to be taking a look for.

You’ve ended my four day lengthy hunt! God
Bless you man. Have a great day. Bye

11. The battery life for the TheraGun is fairly strong.

12. It’s actually very complicated in this full of activity life to listen news
on TV, so I simply use world wide web for that purpose, and obtain the most recent news.

13. wonderful issues altogether, you just gained a new reader.
Any positive?

14. I do not know whether it’s just me or if perhaps everybody else encountering issues with your blog.
It seems like some of the text on your posts are running off the
screen. Can somebody else please provide feedback and let
me know if this is happening to them as well? This could be a issue with my
internet browser because I’ve had this happen previously.
Thanks

15. My family members all the time say that I am wasting my time here
at net, except I know I am getting experience every day by
reading thes good articles or reviews.

16. Nice weblog right here! Additionally your web site rather a lot up very fast!
I desire my website loaded up as quickly as yours lol

17. Thanks for sharing your info. I really appreciate your efforts and I am
waiting for your further write ups thanks once again.

18. Thank you for another wonderful post. The place else may
anyone get that type of information in such a perfect means of writing?
I’ve a presentation subsequent week, and I’m on the look for such information.

19. Cowl something that can be broken with dust cloths.

20. This is how most MM worms spread. Basalt was a galvin.

This site uses Akismet to reduce spam. Learn how your comment data is processed.