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Fair enough. Do you remember how to use integration to calculate the volume between planes and other surfaces?Julio said:Hello, no, it's a cuirosidad. I try to resolve, however, I can not remember how.
I see..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.
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.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...
Huh?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.
That is better, and h'<h is the reason your previous approach didn't work.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!
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.
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.
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.
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.
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.