Why does 2 cones equal a cylinder?

Question

im only talking surface area

Answer 1

Actually, a cone is 1/3 the volume of a cylinder with the same height and the same base diameter.

For the surface, you can get an intuitive understanding by looking at it this way.
A cone's surface varies from the maximum diameter at the base to zero at the tip. The average is thus half way between the base and the tip, where the diameter is half that of the base. Anything you gain moving towards the base is exactly balanced by anything you have moving exactly at the same rate towards the tip.
Hence the surface of the cone would be proportional to 1/2 the base diameter. A cylinder, having the same diameter throughout, will thus have twice the external surface area.

Answer 2

I think this will work, but the cones have to be plastic. There are three phases: I Map the cones to rectangular-based pyramids. II Assemble the pyramids into a square-based prism. With side, s, of the square such that, s^2 = (pi)r^2 and height, h. III Map the square-based prism into the cylinder. In phase I one of the of the rectangular-based pyramids has an s-by-s base and a height h. The other two have an s-by-h base and height s. These are not right pyramids. Rather the apex is above one of the corners of the base. I think the two non-square-based pyramids have to be mirror images of each other. In phase II place the two non-square-based pyramids together so that their s-by-s right triangular faces form an s-by-s square. Then complete the square-based prism with the third pyramid. I don't think any amount of finite cutting of solid cones can turn the cones into a cylinder. This has probably been proven, but it is beyond me. ADDED: Here's another plastic deformation. Take the cone (with apex up) and squash it down into a cylinder 1/3 the height of the cone. In polar cylindrical coordinates transform points on the cone (R, θ, z) by θ' = θ, z' = z/3, R' = Rh/(h-z). The R transformation makes every circular cross-section expand to the radius of the base. I use R because you already defined r as the radius of the cylinder. In cartesian: x' = xh/(h-z), y' = yh/(h-z), z' = z/3. The other two cones undergo these transformations so that they stack Second cone: x' = xh/(h-z), y' = yh/(h-z), z' = z/3 +h/3. Third cone: x' = xh/(h-z), y' = yh/(h-z), z' = z/3 +2h/3. Conceptually, I like the first transformation better, because the transformed cones, though greatly distorted, still have apexes. However, the equations are very bulky.

Answer 3

Well first of all the formula for the surface area of a cylinder (not taking into account the surface area of the round base faces, is

S = 2*pi*r*h, where S is the surface area, r= the radius of the cylinder and h is the height of the cylinder

For two stacked cones the surface area of each cone (once again not taking into acocunt the surface area of the round base faces)

S=4*pi*r*square root of (r^2+(h/2)^2)

So actually the two are not equal.

Answer 4

The surface area of a cone is pir^2 + pirl (l= slant height)
So 2 cones would have a surface area of 2pir^2 +2pirl
The area of a cylinder is 2pir^2 + 2pi r h

So in order for the surface area of two cones to be equal to the area of one cylinder, the slant height of each cone must be equal to the height of the cylinder.

So, in general, your statement is untrue except for the special case where cylinder height = con slant height.

Answer 5

The formula for the volume of a cone can be determined from the volume formula for a cylinder.

It takes the volume of three cones to equal one cylinder.

Answer 6

That's not quite true. At least, in terms of volume.

The volume of a cone is

V = (1/3)pi r^2 h

The volume of a cylinder is

V = pi r^2 h

It follows that 3 identical cones (volume-wise) would equal a cylinder's volume of the same radius, because

(1/3)pi r^2 h + (1/3)pi r^2 h + (1/3)pi r^2 h =
3 [ (1/3)pi r^2 h ] = pi r^2 h

Answer 7

I don't think I understand

Answer 8

No it doesn't. Where did you get that idea?