Consider a spherical wedge that is 1/4 of a ball. Its surface has a flat part (two plane semidisks) and a curved part (a spherical lune).
The areas of these two parts of the wedge's surface are equal; e.g. if the ball has unit radius, then both parts of the surface have area $\pi$.
I find this surprising. Is there a geometrically intuitive direct proof or explanation for it? (Perhaps such a proof would involve, and/or lead to, a smooth animation showing the flat part smoothly morphing into the shape of the curved part, while the area remains constant.)
Similarly, the surface area of a sphere is equal to the area of a disk of twice the radius as the sphere. I'd be interested to see an intuitively clear explanation for this, as well.
The general fact that surprises me here is that disk areas are rational multiples of spherical surface areas. This holds for 2d spherical surface areas in 3d, as in the above examples, but it does not hold if we go up a dimension-- that is, it doesn't hold for 3d spherical surfaces in 4d (since the formula for spherical surface content in 4d involves a factor of $\pi^2$ instead of $\pi$). But it does hold if we go up yet another dimension, to 4d spherical surfaces in 5d. In general, it works for even-dimensional spherical surfaces in odd dimensions, but not vice versa. I've seen this proved using calculus, but I've never understood intuitively why it happens.