Archimedes and the measurement of the circle

The number $\pi$ appears in every formula about the circle — circumference $2\pi r$, area $\pi r^2$ — but what is its actual value? The oldest and most rigorous answer was given by Archimedes of Syracuse (287–212 BC) in the short treatise Measurement of a Circle (in Greek Kýklou métrēsis), three propositions that are a masterpiece of geometric reasoning.

The idea: squeezing the circle between two polygons

A circle is a curve, and measuring it directly is impossible with straightedge and compass. Archimedes got around the obstacle with a simple, powerful idea: enclose the circle between two regular polygons, one inscribed and one circumscribed, whose perimeters we can compute exactly.

The perimeter of the inscribed polygon is shorter than the circumference; that of the circumscribed one is longer. The true length therefore stays “trapped” between the two: $P_{\text{inscribed}} < 2\pi r < P_{\text{circumscribed}}$

The more sides the polygons have, the closer the two perimeters come to each other and the tighter the grip on the unknown value.

The 96-sided polygon

Archimedes did not stop at a few estimates: starting from the regular hexagon he repeatedly doubled the number of sides — $6 \to 12 \to 24 \to 48 \to 96$ — computing at each step, through exact calculations on square roots, the new perimeter. Reaching the $96$-sided polygon he obtained the celebrated double inequality: $3\frac{10}{71} < \pi < 3\frac{1}{7}$

In decimal terms, $3.1408\ldots < \pi < 3.1428\ldots$. The value $\tfrac{22}{7} = 3\tfrac{1}{7}$ is still today the fraction used to approximate $\pi$ in quick calculations (Boyer, Dunham).

Why it matters

Archimedes’ method is the first appearance of an idea that runs through all of mathematics: approximating an elusive quantity by a sequence of estimates from below and from above that squeeze it from both sides. It is the direct ancestor of the concept of limit and of integral calculus, which would be born two thousand years later. In the same spirit, in his work On the Sphere and Cylinder, Archimedes proved that the area of the circle equals that of a triangle whose base is the circumference and whose height is the radius, that is $\pi r^2$ (Katz).

Links

Topics: Euclidean circle
Concepts: Arc · Circle
People: Archimedes