# How do I find the center of mass of a cone?

• schaafde
In summary, the center of mass of a cone can be found by taking a cross-section through the axis and using the Pythagorean theorem to divide the resulting triangle into an upper triangle and a lower trapezoid with equal areas. The center of mass will be at the same spot as the center of the area of the triangle. However, this method only works for solid cones, not hollow ones.

#### schaafde

Im trying to figure out the center of mass of a cone for this research I'm doing. How do I find the center of mass of a cone?

any particular reason you can't research that too?

take a cross-section though the axis. how would you find the center of mass of the resulting triangle?

because I came across something called the Riemann Stieltjes integral and it is making no sense to me

schaafde said:
because I came across something called the Riemann Stieltjes integral and it is making no sense to me

So did my question not make sense to you? Finding the center of gravity of a cone is exceedingly trivial and you do not need anything but the Pythagorean theorem. You certainly don't need any calculus, although I'm sure there are lots of hard ways to do it if you WANT to do it a hard way.

No, I never said that. To determine the centroid of a general figure, you certainly need Calculus. But, you can then use a formula for a given type of figure- there is a standard formula for the centroid of a cone. Perhaps the OP could just look that up.

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HallsofIvy said:
To determine the centroid of a general figure, you certainly need Calculus.

agreed

phinds,
could you elaborate on your method of using only the Pythagorean theorem? I'm not getting it.

The center of gravity of a cone is at the exact same spot as the center of the area of a triangle created if you pass a plane through the axis of the cone. Divide that triangle into an upper triangle (I'm thinking of the cone pointing downward) and a lower trapazoid, with the areas of the two being equal. Pythagorous (and an understanding of similar triangles) will give you the rest.

I think you are right if you are considering a hollow cone. I was assuming the OP asked about a solid cone, but of course I might be wrong.

For me, it is not very intuitive that the cross-sectional triangle trick will work if a hollow cone is considered. How did you come to that conclusion?

Guffel said:
I think you are right if you are considering a hollow cone. I was assuming the OP asked about a solid cone, but of course I might be wrong.

For me, it is not very intuitive that the cross-sectional triangle trick will work if a hollow cone is considered. How did you come to that conclusion?

I have no idea why you think the cone should be hollow for this to work; I'm assuming a solid cone. If the upper triangle has the same area as the lower trapezoid then will not the resulting rotated figures have the same volumes?

OH ... oops, maybe they don't

phinds said:
I have no idea why you think the cone should be hollow for this to work; I'm assuming a solid cone. If the upper triangle has the same area as the lower trapezoid then will not the resulting rotated figures have the same volumes?

OH ... oops, maybe they don't

LATER: I checked it out and it does work out correctly in this case.

phinds said:
The center of gravity of a cone is at the exact same spot as the center of the area of a triangle created if you pass a plane through the axis of the cone. Divide that triangle into an upper triangle (I'm thinking of the cone pointing downward) and a lower trapazoid, with the areas of the two being equal. Pythagorous (and an understanding of similar triangles) will give you the rest.

The median of an isosceles triangle is H/3 while the CoM of a cone is at H/4. I'm not quite sure how your method works out.

Yuqing said:
The median of an isosceles triangle is H/3 while the CoM of a cone is at H/4. I'm not quite sure how your method works out.

I don't know what the median has to do with anything and the CoM of a cone, if I have it right is at H/(1-SQRT(2)) assuming the cone is pointed up

phinds said:
I don't know what the median has to do with anything and the CoM of a cone, if I have it right is at H/(1-SQRT(2)) assuming the cone is pointed up

But the CoM of the cone is at H/4, assuming the cone is pointed up. There seems to be some miscommunication here. Edit: 1-sqrt(2) isn't even positive.

Yuqing said:
But the CoM of the cone is at H/4, assuming the cone is pointed up. There seems to be some miscommunication here. Edit: 1-sqrt(2) isn't even positive.

Yeah, sorry about that, got rushed. Meant to say 1-(1/sqrt(2)), or in other words, .707.

I'm running a "stack of disks" computer program just now but seem to have something messed up since instead of .707, or the .75 you say is the right answer, I'm getting .794

Is the center of mass of a cone in a different place than the center of area of a triangle created by intersecting a plane through the axis of the cone? If it is, then I've got it wrong 'cause that's what I'm basing it on.

phinds said:
Yeah, sorry about that, got rushed. Meant to say 1-(1/sqrt(2)), or in other words, .707.

I'm running a "stack of disks" computer program just now but seem to have something messed up since instead of .707, or the .75 you say is the right answer, I'm getting .794

Is the center of mass of a cone in a different place than the center of area of a triangle created by intersecting a plane through the axis of the cone? If it is, then I've got it wrong 'cause that's what I'm basing it on.

Was your assumption that the CoM of the cone is vertically at the same location of the CoM of the triangle? If so then that's not right. When you rotate the triangle, more weight is prescribed to the larger end, so intuitively the CoM should move towards the larger end.

phinds said:
Yeah, sorry about that, got rushed. Meant to say 1-(1/sqrt(2)), or in other words, .707.

I'm running a "stack of disks" computer program just now but seem to have something messed up since instead of .707, or the .75 you say is the right answer, I'm getting .794

Is the center of mass of a cone in a different place than the center of area of a triangle created by intersecting a plane through the axis of the cone? If it is, then I've got it wrong 'cause that's what I'm basing it on.

The com of the section area as almost nothing to do with the com of the solid.
For example consider two cubes: the first with volume 1 and the other with volume 8.
The com of the section is at 1/5 the distance from the bigger one; the real com is at 1/9.

If you write the integral you will see that the com of the cone is related to that of the 2d figure which has two parabolas as borders.

OK, I clearly got it wrong. Sorry about wasting everybody's time on this. I thought I had a simple solution but it turns out I just have a simple mind.

## 1. How is the center of mass of a cone defined?

The center of mass of a cone is the point at which the entire weight of the cone can be considered to act, making it the balance point of the cone.

## 2. What is the formula for finding the center of mass of a cone?

The formula for finding the center of mass of a cone is (1/4) x (height of the cone) x (radius of the base squared) / (height of the cone) + (1/2) x (height of the cone). This can also be written as (3 x height of the cone) / (4 x height of the cone) + (1/2) x (height of the cone).

## 3. Is the center of mass of a cone always located on the central axis?

Yes, the center of mass of a cone will always be located on the central axis, as it is the point at which the weight of the cone is evenly distributed.

## 4. Can the center of mass of a cone be outside of the cone?

No, the center of mass of a cone will always be located within the boundaries of the cone itself.

## 5. How does the shape and dimensions of a cone affect its center of mass?

The shape and dimensions of a cone will affect the location of its center of mass. A taller and narrower cone will have a higher center of mass, while a shorter and wider cone will have a lower center of mass. Additionally, a cone with a more pointed tip will have a higher center of mass than a cone with a flatter tip.