# Moment of inertia of a regular triangle

• Who_w
In summary: It's always interesting to learn more about how others think. In summary, the first way does not work and the second way gets you a different answer.
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?

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• q1.PNG
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• q2.PNG
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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.

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Last edited:
Who_w said:
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
haruspex said:
(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!

Who_w said:
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!

## 1. What is the formula for calculating the moment of inertia of a regular triangle?

The formula for calculating the moment of inertia of a regular triangle is (1/36) x m x a^2, where m is the mass of the triangle and a is the length of one of its sides.

## 2. How does the moment of inertia of a regular triangle compare to that of a square or circle?

The moment of inertia of a regular triangle is smaller than that of a square or circle with the same mass and dimensions. This is because a triangle has less mass concentrated at the center compared to a square or circle, resulting in a smaller moment of inertia.

## 3. How is the moment of inertia of a regular triangle affected by its orientation?

The moment of inertia of a regular triangle is affected by its orientation in relation to the axis of rotation. It is highest when the triangle is rotated about an axis passing through its centroid and perpendicular to its plane, and lowest when rotated about an axis parallel to one of its sides.

## 4. Can the moment of inertia of a regular triangle be negative?

No, the moment of inertia of a regular triangle cannot be negative. It is always a positive value, representing the resistance of the triangle to rotational motion.

## 5. How is the moment of inertia of a regular triangle used in real-world applications?

The moment of inertia of a regular triangle is an important concept in physics and engineering, and is used in various real-world applications such as calculating the stability of structures and predicting the behavior of rotating objects.

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