Can someone draw a picture of this situation? I can't make sense of it

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In summary, the conversation explains that the sphere's area is four times that of a flat circle with the same perimeter due to the intersection of a cylinder and sphere at the equator, with the area of their intersecting planes being equal regardless of their location.
  • #1
The following was given as an intuitive explanation of understanding why a sphere's area is four times the area of the flat circle sharing a perimeter with a greater circle on the sphere. I don't understand spatially how the cylinder and sphere are oriented. Please help.

Let Z be a cylinder of height 2r touching the sphere Sr along the equator θ=0. Consider now a thin plate orthogonal to the z-axis having a thickness Δz≪r. It intersects Sr at a certain geographical latitude θ in a nonplanar annulus of radius ρ=rcosθ and width Δs=Δz/cosθ, and it intersects Z in a cylinder of height Δz. Both these "annuli" have the same area 2πrΔz. As this is true for any such plate it follows that the total area of the sphere Sr is the same as the total area of Z, namely 4πr2.
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  • #2
Attached is a .png file of what they are talking about, also a Mathematica notebook that generated it. The red sphere and the blue cylinder intersect at the equator, shown in black. They are saying that the area of intersection of the green plane with the blue cylinder is equal to the area of intersection of the green plane with the red sphere, no matter where the green plane is located.


  • Cylinder.nb
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1. How detailed should the picture be?

The level of detail in the picture will depend on the complexity of the situation and what information you are trying to convey. It is important to communicate with the person drawing the picture to ensure that the level of detail is appropriate.

2. Can the picture be hand-drawn or does it need to be digital?

This will depend on personal preference and the available resources. Hand-drawn pictures can be just as effective as digital ones, as long as they accurately convey the information. However, digital pictures may be easier to edit and share.

3. How long will it take to draw the picture?

The time it takes to draw the picture will depend on the complexity of the situation and the skill of the person drawing. It is important to allow enough time for the picture to be drawn accurately and effectively.

4. Do I need to provide any specific instructions for the picture?

It is helpful to provide some guidelines or instructions for the picture, such as the main elements that should be included or any key details that need to be highlighted. However, it is also important to give the person drawing the picture creative freedom to best convey the situation.

5. How will the picture be used?

The purpose of the picture will determine how it will be used. It could be used for communication, documentation, or visualization. It is important to clarify the purpose beforehand so that the picture can be drawn accordingly.

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