Matpic
mathematics, in pictures
Series

Circle area — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · geometry
  2. 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.

    part 2 · geometry
  3. A hemisphere and a cylinder with a cone taken out, sliced at one height. Two solids drawn in profile — a hemisphere, and a cylinder with a cone removed — each cut at the same height, with the disc and the annulus the cut produces marked and their equal areas given.

    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.

    part 3 · geometry

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