[00:00] exactly 4 times the area of a circle with the same radius. Archimedes had a very beautiful proof that this surface area is the same as the area of
[00:13] a cylinder that encloses that sphere, if you disregard the circular caps of that cylinder. The idea is that if you shine some light from the z-axis perpendicular to that axis, then if you compare the area of a small rectangle drawn on that sphere to the area
[00:27] of the shadow cast on that cylinder, those two areas turn out to be the same. This isn't obvious, but when you work it out, that shadow is a copy of the original little rectangle, but squished down in one direction and stretched out in another.
[00:41] When you analyze the relevant geometry, the two effects actually cancel out perfectly. and the cylinder as a sum of all of those shadows, we can infer that the surface area of the sphere is the same as the area of that cylinder.
[00:56] where one side length corresponds to the circumference, 2 pi r, and the other corresponds to the height of the sphere, 2 times its radius. And if you're also familiar with the trick of showing a circle's area,
[01:12] by unwrapping it into a triangle, one whose height is r and whose base is 2 pi times r, you can see how four of those unwrapped circles fit perfectly into this unwrapped shape.