|Sep11-12, 06:51 PM||#1|
Can someone draw a picture of this situation? I can't make sense of it!
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.
|Sep12-12, 06:25 AM||#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.
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