Squeezing π CalcLens

A circle of radius 1 has area π. Trap it between a polygon drawn inside and a polygon drawn outside, then add sides. Neither polygon is a circle, but together they leave the circle nowhere to hide — which is how Archimedes found π, and it is the squeeze theorem two thousand years early.

Archimedes' doubling

He started with a hexagon and doubled, by hand, to 96 sides
sidesinside outsidegap

Squeezing π

The circle has radius 1, so its area is π. The polygon drawn inside it is too small; the polygon drawn outside is too big. Add sides and the two close in.

The two areas

For a regular n-gon in a unit circle, , and for the one around it, . Neither is ever equal to π — but every value of n gives a genuine pair of bounds, and the gap between them goes to zero.

Why this is the squeeze theorem

Two sequences with the same limit, and the thing you care about trapped between them. That is the whole hypothesis, and Archimedes was using it in the third century BC — with 96 sides, by hand, no decimal notation and no trigonometry tables, to get .

A footnote for accuracy: he squeezed the circumference between the two perimeters, not the area between the two areas. Both work, and both give π. The area version is the one shown here because it is the one that leads to sin(x)/x below.

Where sin(x)/x comes from

Look at one wedge and cut it in half. With the half-wedge gives three areas — a triangle inside, the sector, and a triangle outside — and reading them off is exactly .

Divide by sin x and turn it over, and you get . Both ends go to 1, so the thing between them does too. The limit we need for the derivative of sin is the same picture as Archimedes' π — it is this figure, read at the level of one wedge instead of the whole circle.

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