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## Main Question or Discussion Point

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 S

Let Z be a cylinder of height 2r touching the sphere S

_{r}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 S_{r}is the same as the total area of Z, namely 4πr^{2}.