Free Moment of Inertia Calculator / Centroid Calculator

Calculate Moment of Inertia, Centroid, Section Modulus of Multiple Shapes

This free SkyCiv's Free Moment of Inertia Calculator is able to accurately and quickly calculate the moment of inertia among other section properties.

How to use this Moment of Inertia Calculator?

To start using the free centroid calculator to calculate moment of inertia, simply choose the cross section you want to evaluate from the drop down list, then enter in the dimensions and click Calculate. The results for centroid, moment of inertia, statical moment section modulus and torsion constant will display on your right.

Find the moment of inertia with ease

This free multi-purpose calculator is taken from our full suite Structural Analysis Software. It allows you to:

  • Calculate the Moment of Inertia (I) of a beam section (Second Moment of Area)
  • Centroid Calculator used to calculate the Centroid (C) in the X and Y axis of a beam section
  • Calculate the First moment of area (Statical Moment of Inertia) (Q) of a beam section (First Moment of Area)
  • A Section Modulus Calculator to calculate the Section Modulus (Z) of a beam section
  • Calculate the Torsion Constant (J) of a beam section

For more professional project needs:

To access the full functionalities of SkyCiv Structural Analysis Software, check out our professional plans. We have flexible plan options that tailor to different needs.



Notation Value Unit
A - -
Iz - -
Iy - -
Izp - -
Iyp - -
αp - -
Cz - -
Cy - -
Qz - -
Qy - -
Sz - -
Sy - -
J - -

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Hand Calculations

SkyCiv shows you how to calculate the moment of inertia by showing the step-by-step moment of inertia equations and calculations... The also show centroid and area calculations.

Custom and Built-Up Sections

SkyCiv Section Builder uses FEA to calculate key section properties, allowing you to model anything you need. Simply upgrade and get access to powerful software.

Integrated with Analysis Software

The SkyCiv Section Builder is fully integrated with all SkyCiv Analysis Software including SkyCiv Beam and SkyCiv Structural 3D - so you can analyze your section under loads.

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About the Moment of Inertia Calculator

This simple, easy-to-use moment of inertia calculator will find moment of inertia for a circle, rectangle, hollow rectangular section (HSS), hollow circular section, triangle, I-Beam, T-Beam, L-Sections (angles) and channel sections, as well as centroid, section modulus and many more results.

You can solve up to three sections before you're required to sign up for a free account - which also gives you access to more software and results.

Moment of Inertia

For instance, if you're looking how to calculate the moment of inertia of a rectangle you can use the tool above simply by selecting rectangle from the drop down list then entering some dimensions for height and width (e.g. 100, 200). After clicking "Calculate", the tool will calculate the moment of inertia. Our paid version will show the full hand calculations of how the tool got to this result. These detailed step-by-step instructions help for you to follow the calculations and learn how to calculate the moment of inertia. Alternatively, we have a tutorial on how to find moment of inertia.

Section Modulus Calculator

The calculator will also produce results like section modulus (Z) and the statical moment of inertia (Q) for your sections. Finally, the result for Torsion Constant (J) will also display using this tool. The SkyCiv Section Builder will also produce these results accurately using FEA analysis of the cross section. So if you need more results, or to add custom shapes - please check out our extra functionality of the SkyCiv Section Builder.

More Tools Available

SkyCiv also offers other tools such as I beam size tool and free structural design software. The dynamic section drawer will also show you a graphical representation of your beam section. So if you want to calculate the moment of inertia of a circle, moment of inertia of a rectangle or any other shapes, feel free to use the below software or our all-inclusive SkyCiv Section Builder.

About SkyCiv

SkyCiv offers a wide range of Cloud Structural Analysis and Design Software for engineers. As a constantly evolving tech company, we're committed to innovating and challenging existing workflows to save engineers time in their work processes and designs.

If you are an engineering professional, you might benefit more by using our cloud based 3D Structural Analysis Software beyond this calculator. This software includes integrated design modules such as AISC, ACI, AS, Eurocode and CSA. Check out our professional packages here:

3D Structural Analysis Software

Responses (47)

  1. Adham Nadim
    October 18, 2017 at 9:29 pm · Reply

    What deflection will a standard I beam 200 mm 10 meters long free support have with a free span of 8 meters in the middle with a 4000 kg distributed live load over the 8 meters

  2. Kendy Valeta
    May 30, 2017 at 6:41 am · Reply

    No funciona la viga canal

  3. David Wimberley
    March 23, 2017 at 4:16 am · Reply

    I’m a first time user and fan of what you’re doing, but I have a constructive suggestion. It seems to me intrinsically confusing to have two rectangular cross-sections in your example, one of which is higher than it is wide, and the other wider than it is high. It would be much less confusing to be able to input your data into a single rectangle. Thanks for hearing me out.

    • Paul
      March 23, 2017 at 8:14 am · Reply

      Hi David,

      I completely agree. This is why in our new section builder you just need to specify the outer width and depth with a thickness. If you sign up for free account, you can access it by clicking ‘SB’ in the left menu:

      • David Wimberley
        March 24, 2017 at 4:30 am · Reply

        OK. Did so. Thanks.

    January 4, 2017 at 7:25 am · Reply

    can anyone differentiate between moment of inertia of two rectangular beams, with and without holes over it.

  5. Anitha G
    November 18, 2016 at 4:54 pm · Reply

    am designing triangular cantilever beam, how to calculate stress, displacement, moment inertia, and theoretical i have to prove, plz suggest me and give some material to understanding.

  6. Shaun Murrin
    November 7, 2016 at 7:01 am · Reply

    This is a tutorial question I’m struggling a bit which axis in to vase the I value in..”Based on calculations it has been determined that a universal steel column must have a minimum value of I of 38748 cm4. Select a column suitable for this scenario showing clearly why you have selected that particular column” .

  7. Chris Webb
    September 26, 2016 at 3:42 pm · Reply

    I hate the moor

  8. jamsheed sajid
    June 9, 2016 at 6:20 pm · Reply

    wow a nice and great work done by you. being a mechanical engineering student i think now i have found all what i need through out degree. thanks a lot for such a great work. i salute you

  9. Gokul Kgs
    June 8, 2016 at 5:19 pm · Reply

    Can anyone help me to find out the moment of inertia for this type of I-Profile.
    I need a little detail explanation.

    • Paul
      June 8, 2016 at 5:31 pm · Reply

      Hi Gokul. Wow this is a complex one! The first step is always finding the centroid of the area as that will tell you where the neutral axes are. Then you need to divide the area into different sections. Both of these steps are explained in our tutorials:

      This is a complex section though – it will probably take 1+ hours by hand. Hopefully in future we can add custom section shapes in future.

      • Gokul
        June 8, 2016 at 6:52 pm · Reply

        Thanks for your reply.
        i just want to know that how to consider the curves present in it. Is there any way to find easily like software……..

        • mecheng
          June 24, 2016 at 7:42 pm · Reply

          Maybe a bit too late to be helpful, but I’d just sketch it in Solidworks or similar and use the built in ‘evaluate section’ function. Any proper CAD system should have this function. Google will help you a long way on how to do it 😉

        • Muthamizhselvan Vijayan
          August 16, 2016 at 8:36 pm · Reply

          Hi gokul, better you extrude this profile in creo parametric and go for mass properties..

    • Pape Modou Fall
      August 7, 2016 at 11:29 pm · Reply

      you can use autocad logiciel

      1-at first you drax this section
      2-you click left “Region” and selection all the section
      3-finaly , you click left again for select “Propmeca”, you select your region

      and you have ( Iy, Iz, Go, A, P)
      if you draw it at dwg, I can do it for you

    • HollywoodF1
      March 9, 2017 at 4:20 am · Reply

      Check with the Aluminum Design Manual (I’m assuming this is Aluminum.) The section properties are limited by the strength of the elements of the section. SAP 2000 has a tool where it will calculate the section properties of shapes you import from AutoCAD.

      June 3, 2017 at 7:20 pm · Reply

      Hi Gokul. Try any 3D software help. You could solve this within 5 to 10 min for sketching but you could get exact results. As per my suggestion use Creo Software.

    • Sam Carigliano
      June 22, 2017 at 3:10 am · Reply

      Hey Gokul,
      Sam from SkyCiv here. With our Section Builder Software (under Student or Professional account) you can solve complex structures like your attached (this dimensions aren’t the same as yours but you get the idea):

    • Sam Carigliano
      June 22, 2017 at 3:19 am · Reply

      Hey Gokul,
      Sam from SkyCiv here. With our Section Builder Software (under Student or Professional account) you can solve complex structures like your attached (this dimensions aren’t the same as yours but you get the idea):

  10. Raghda Al Rashii
    May 16, 2016 at 4:22 am · Reply

    I don’t get how did you find Iz?

  11. lifestyletips2
    February 25, 2016 at 3:09 pm · Reply

    It’s not easy to find such perfectly written information on this topic,great work,thanks for sharing….!

  12. Roland
    February 5, 2016 at 12:20 am · Reply

    Why does it give Iz and Iy rotated for a L-beam? What if I want the area moment of inertia about the vertical and horizontal axes?

    • Paul
      February 5, 2016 at 11:57 am · Reply

      L-sections and some channel sections are asymmetric in both the horizontal and vertical axes. So when a load is applied the member actually twists (see the image from SkyCiv Structural 3D renderer – only a vertical load is applied yet there is displacement in two directions). The Iz and Iy values shown are about the principal (rotated) axes. Perhaps in the future we can provide the values about the vertical and horizontal axes too. Thanks for the suggestion.

      • travis
        February 4, 2017 at 2:01 am · Reply

        beautiful pic! (:

        • travis
          February 4, 2017 at 2:02 am · Reply


  13. Al
    November 25, 2015 at 8:43 pm · Reply

    section modulus in the ‘Z’ and ‘Y’ axis should not be the same value?

    • Paul
      November 26, 2015 at 8:13 am · Reply

      Which section are you referring too so we can check it? If it is symmetric then it would be the same value.

  14. Manuel Sierra
    November 18, 2015 at 9:46 am · Reply

    Muy útil, gracias mil

  15. Guilherme Henrique
    November 6, 2015 at 4:22 pm · Reply

    this is very useful, but it’s missing a geometric composition tool. Thanks anyway.

  16. Zia Ul Islam
    August 26, 2015 at 11:43 pm · Reply

    Hey Paul, Thank you so much. Can you please put up ” Z ” shaped cross sections in there as well ? Would be greatly appreciated :)

    • Paul
      August 27, 2015 at 2:54 pm · Reply

      Hi Zia, I really hope in the near future we can add many many more sections :) including the Z-section!

  17. Tarek J. Ammar
    July 31, 2015 at 11:29 pm · Reply

    Would be great if you can include calculations for notched beams ( For connections’ purposes)

    • Paul
      August 1, 2015 at 3:09 pm · Reply

      Hi Tarek. Thanks for the feedback although I’m not 100% sure what you mean – could you please elaborate or provide an example? Thanks.

      • Tarek J. Ammar
        August 3, 2015 at 8:42 am · Reply

        Anytime. Well mostly in designing connections for beams-to-beams/columns you’re required to notch same-sized beams at the end so they can be connected to the web so I section becomes a notched T section. Furthermore its necessary to find the moment of inertia and stresses onto that beam

  18. Bak
    July 23, 2015 at 12:35 am · Reply

    hi. what about a reversed t beam ? Thanks.

    • Paul
      July 23, 2015 at 10:50 am · Reply

      Hi Bak. If you invert the T-beam the results would be identical except for the values of the Centroid since they are always measured from the bottom left of the section. Hope that helps!

  19. Vincent
    July 17, 2015 at 7:42 am · Reply

    Is it possible to get the equation for Ix and Iy for the T-Beam?

    • Paul
      July 17, 2015 at 10:22 am · Reply

      Hi Vincent. We’ll be adding the formulae for most results in the near future. Sorry but the T-beam is very complex to type out so you’re going to have to wait. Sorry about that.

  20. lir
    June 29, 2015 at 4:15 pm · Reply

    The origion of axis system should be at Center of mass, otherwise results are not correct.

    • Paul
      July 1, 2015 at 7:57 pm · Reply

      Thanks for letting us know. This has been fixed.

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