Moment of inertia of a regular triangle

[Who_w]

Homework Statement

Calculate the moment of inertia of a triangle relatively OY

Relevant Equations

Given the height of the triangle

Please, I need help! I need to calculate the moment of inertia of a triangle relatively OY. I have an idea to split my triangle into rods and use Huygens-Steiner theorem, but after discussed this exercise with my friend, I have a question: which of these splits are right (picture 1 and 2)? Or maybe my idea is wrong?

 

[haruspex]

Each can work, but it would be more natural to use the one where the strip is parallel to the axis.

 

[Who_w]

I found a mistake in the first split, thanks a lot! I would be very grateful if you tell me where the error is (attached the file). With another split, I got a different answer. I solved it 3 times in this way, but did not find an error.

 

[haruspex]

I found a mistake in the first split, thanks a lot! I would be very grateful if you tell me where the error is (attached the file). With another split, I got a different answer. I solved it 3 times in this way, but did not find an error.

(l-x)dx? Both of those distances are parallel to the x axis. It is not the area of a rectangle in the figure.
Also, you seem to be taking a strip parallel to the x axis, which is not what I recommended. Are you just checking it can be done both ways?

 

[Who_w]

(l-x)dx? Both of those distances are parallel to the x axis. It is not the area of a rectangle in the figure.
Also, you seem to be taking a strip parallel to the x axis, which is not what I recommended. Are you just checking it can be done both ways?

Oh! I understood my mistake! Thank you a lot :)
Yes, I'm tried to check both ways. The first way worked (strip parallel to the y axis), but it was really interesting for me why I couldn't to solve it the second way.
Thanks again!

 

[haruspex]

Oh! I understood my mistake! Thank you a lot :)
Yes, I'm tried to check both ways. The first way worked (strip parallel to the y axis), but it was really interesting for me why I couldn't to solve it the second way.
Thanks again!

I commend your inquisitiveness.