Problem about a cone and plane cutting it

In summary, the angle between the plane and the base of the cone should be ##h-\frac {h} { \sqrt[3] 2}## in order for the plane to intersect the cone in two equal parts.
  • #1
Julio
2
0
It has a straight cone and a plane that cuts it as shown in the image. What angle should the plane form with the base of the cone ?, so that the plane cuts the cone in 2 parts of equal volume.

Cono.jpg
 

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  • #2
Welcome to the PF. Is this question for schoolwork?
 
  • #3
Hello, no, it's a cuirosidad. I try to resolve, however, I can not remember how.
 
  • #4
Julio said:
Hello, no, it's a cuirosidad. I try to resolve, however, I can not remember how.
Fair enough. Do you remember how to use integration to calculate the volume between planes and other surfaces?
 
  • #5
The problem is ill-posed. It has an infinitude of solutions as it's depending on the height of the plane above ground it cuts cone rather than the angle.

For example if my quick calculations are correct, for angle 0 degrees of plane parallel to ground a solution would be: ## h-\frac {h} { \sqrt[3] 2} ##.
 
  • #6
Based on the image I guess the cutting plane should intersect the floor of the cone at the outer edge of the base.
The angle will still depend on the height/radius ratio but that dependence is trivial.
 
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  • #7
mfb said:
Based on the image I guess the cutting plane should intersect the floor of the cone at the outer edge of the base.
The angle will still depend on the height/radius ratio but that dependence is trivial.
I see..

Still it depends on the orientation of the plane then. Is the line of cut of the tilted plane with the z=0 plane the x=0 line, the y=x line? there are y=nx solutions...

For the simplest case, here's a reference: http://mathworld.wolfram.com/EllipticCone.html
 
Last edited:
  • #8
Rada Demorn said:
Still it depends on the orientation of the plane then. Is the line of cut of the tilted plane with the z=0 plane the x=0 line, the y=x line? there are y=nx solutions...
One assumes the intersection of the tilted plane with the z=0 plane forms a line which is expected to be tangent to the base of the cone. And one assumes a right circular cone.
 
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  • #9
So. find the major and minor semi-axes a,b of the ellipse as functions of the angle of tilted plane and pluck them in ## a.b = r^2 /2 ##

Good luck with the calculations!
 
  • #10
OP asked about the volumes not about the area of the cut. The area of the cut will be larger than half the base.
 
  • #11
mfb said:
OP asked about the volumes not about the area of the cut. The area of the cut will be larger than half the base.
Huh?

Volume of elliptic cone is half the volume of original cone. So, ##
π.a.b.h/3 = π.h.r^2 /6 =>
a.b = r^2 /2
##

Where did you see an area?
 
  • #12
On second thought ##

a.b = r^2 /2 ## is not the right formula to be used...

The definition of an elliptic cone given in http://mathworld.wolfram.com/EllipticCone.html is not applicable in this case. What we are having here is a right circular cone cut by an oblique plane.

So in ##
π.a.b.h'/3 = π.h.r^2 /6 ## I would be using h' instead where h' is the height of the perpendicular drawn from the apex of the cone to the oblique cutting plane.
Again I am not certain about this so you must wait until I verify this.
 
  • #13
Yes, ##
π.a.b.h'/3 = π.h.r^2 /6 ## where h' is the height of the perpendicular drawn from the apex of the cone to the oblique cutting plane and h,r the height and radius of the original cone is the correct formula to be used!
 
  • #14
The upper volume is not an elliptic cone although it happens to have the same volume as one.
Rada Demorn said:
Yes, ##
π.a.b.h'/3 = π.h.r^2 /6 ## where h' is the height of the perpendicular drawn from the apex of the cone to the oblique cutting plane and h,r the height and radius of the original cone is the correct formula to be used!
That is better, and h'<h is the reason your previous approach didn't work.

Define ##L=\sqrt{h^2 + r^2/4}##, the distance between apex and the circumference of the intersection between cone and horizontal plane. Define ##\alpha## to be the angle between cone and horizontal surface and ##\gamma## to be the angle between cut plane and horizontal surface. Define ##\beta= \pi - 2 \alpha## (angle at the apex). Then ##h' = L \sin(\alpha - \gamma)##.
From the law of sines we get ##\frac{L}{\sin(\alpha+\gamma)} = \frac{a}{\sin(\beta)}##.
Now we just have to find b.
 
  • #15
Can you move the plane?(Since I didn’t see any post above says that you can,t) if so,can you use the plane to cut the cone vertically in half?(This is quite ridiculous since some actual attempts have been mentioned above)
 

What is a cone and how is it related to a plane?

A cone is a three-dimensional geometric shape that has a circular base and a curved surface that tapers to a point. It is related to a plane because a plane can intersect a cone at different angles, resulting in different cross-sectional shapes.

What are the different types of plane cuts that can be made on a cone?

There are three main types of plane cuts that can be made on a cone: a parallel cut, an oblique cut, and a perpendicular cut. A parallel cut results in a circle, an oblique cut results in an ellipse, and a perpendicular cut results in a triangle.

How does the angle of the plane affect the resulting cross-sectional shape of a cone?

The angle of the plane in relation to the cone's axis determines the resulting cross-sectional shape. A parallel plane will result in a circle regardless of its position, while an oblique plane will result in an ellipse with varying lengths and widths depending on the angle. A perpendicular plane will result in a triangle with its base being the diameter of the cone's base.

How is the volume of a cone affected by a plane cutting it?

The volume of a cone is not affected by a plane cutting it. The volume of a cone is calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height. The cross-sectional shape created by the plane does not affect these values and therefore does not affect the volume.

What real-world applications use the concept of cutting cones with planes?

The concept of cutting cones with planes has various real-world applications, such as in manufacturing to create different shapes for objects, in architecture to design unique structures, and in mathematics to study the relationship between different geometric shapes. It is also used in engineering for creating efficient designs for structures such as bridges and towers.

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