Circle area — the series
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A circle unrolled into a triangle
Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.
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Pinned between two sequences
The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.
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The slice that has to match
The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.
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A band of a sphere is a band of its cylinder
Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.
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The ball that is largest in five dimensions
A disc of radius one has area π, a ball of radius one volume 4π/3, and in each further dimension the unit ball grows — until five, where its volume is 5.264, after which it shrinks towards nothing. The recursion that shows it is the ring dissection of the disc, done one dimension at a time, and the shrinking is not the ball getting small: it is almost all of a high-dimensional cube lying outside the ball, and almost all of the ball lying in a thin rind at its surface.
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The five crescents that can be squared
The circle cannot be squared, but some regions bounded by circular arcs can: Hippocrates of Chios found a crescent with exactly the area of a triangle around 440 BC. The trick is to make two multiples of π cancel, and it works for a crescent whose two arcs turn through angles in the ratio m to n exactly when an angle defined by a polynomial can be drawn with ruler and compass. That happens for five ratios and no others.
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The error that predicts itself
Archimedes' polygons close on π slowly — six doublings of the sides for two decimals. But the error does not fall at random. Each doubling adds almost exactly a quarter of the area the doubling before it added, so the rest of the slivers can be summed before they are drawn. Liu Hui saw the quarter in 263; Snell and Huygens turned it into bounds; and the 96-gon that gave Archimedes two decimals gives six once its error is known.