Concept

Circle area

The space enclosed by a circle, which is half its circumference times its radius and therefore pi times the square of the radius. The half-base-times-height form is the one the ring dissection establishes, and it makes no reference to pi at all — the constant arrives only when the circumference is written out.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

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.

geometry · Circle area

A circle trapped between two 12-sided polygons. A circle with a regular polygon of 12 sides inscribed in it and another circumscribed about it, beside a table of the bounds on pi obtained by doubling the side count.

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.

geometry · Circle area

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.

geometry · Circle area

Named alongside it

The objects these essays reach for when they reach for this one.

Archimedes Limit Method of exhaustion Area Dissection Pi Approximation Convergence rate

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