Archimedes Traps Pi

Long before calculators, Archimedes surrounded a circle by two polygons. One perimeter is too small, one is too large, and pi is forced to live between them.

University of Michigan Robotics

Number of polygon sides 12

Archimedes' bounds

Midpoint estimate

Error no larger than

The blue polygon sits inside the circle, so its perimeter is too small. The dark polygon sits outside the circle, so its perimeter is too large.

As the number of sides increases, the two perimeters squeeze pi from below and above. For very large values, the picture is visually capped because the polygons are already indistinguishable from the circle; the numerical bounds still use the displayed side count.

Circle with diameter 1

Inscribed polygon: lower bound

Circumscribed polygon: upper bound