Centroids Simplifying by symmetry

[SpartanG345]

If you have a 3d shape how do you simplify the problem using symmetry.

eg with a sphere is symmetric along any axis therefore the centroid must be in the middle of it

eg2 A cone sitting on the xy plane, where the pointy bit is points up the z axis.
-nb it is sitting on point (0,0,0) where the circular crossection lines on the xy plane at z = 0
with a centre at (0,0)

you can intuitively see that the centroid must have an x coordinate of 0 and a y coordinate of 0 and there some random number for the z coordinate.

where my understanding of symmetry is if a shape can be rotated about an axis and look the same then it is symmetric... this many not the be official definition though

it seems to me that if a shape is symmetric across an axis, only a distance along that axis will need to be calculated to get the centroid. Is this right?

 

[tiny-tim]

Hi SpartanG345!

… where my understanding of symmetry is if a shape can be rotated about an axis and look the same then it is symmetric... this many not the be official definition though

It's good enough!

it seems to me that if a shape is symmetric across an axis, only a distance along that axis will need to be calculated to get the centroid. Is this right?

("symmetric about an axis")

Yup, that's completely right!

 

[Mene]

Yes, you are correct. Symmetry can be a very useful tool for simplifying problems in science, particularly in geometry and physics. In the case of finding the centroid of a 3D shape, if the shape is symmetric along any axis, then the centroid must lie on that axis. This means that we only need to calculate the distance along that axis to determine the centroid, rather than having to calculate the centroid for the entire shape.

In the example of a sphere, as you mentioned, it is symmetric along any axis, so the centroid must be located at the center of the sphere. This makes it much simpler to calculate the centroid, as we only need to know the radius of the sphere.

Similarly, in the example of a cone, if it is symmetric along the z-axis, then the centroid must have an x and y coordinate of 0, and we only need to calculate the z coordinate.

Your understanding of symmetry is correct - if a shape can be rotated about an axis and still look the same, then it is symmetric. This can be a very useful concept in scientific research, as it allows us to simplify complex problems and focus on specific aspects of a shape or object.