Archimedes used polygons , specifically regular polygons inscribed in and circumscribed around a circle, starting from a hexagon and refining up to a 96‑sided polygon.

Quick Scoop: What shape did Archimedes use to approximate pi?

Archimedes didn’t use a single “fancy” curve to approximate pi – he used straight‑edged shapes: regular polygons.

  • He began with a regular hexagon inside a circle.
  • Then he doubled the number of sides repeatedly: 6, 12, 24, 48, and finally 96‑sided polygons.
  • He used both:
    • a polygon inscribed in the circle (inside, giving an underestimate), and
    • a polygon circumscribed around the circle (outside, giving an overestimate).

By squeezing the circle between these two polygons, their perimeters closed in around the true circumference, trapping pi between two rational bounds.

Mini story: How his idea works

Imagine drawing a circle, then fitting a neat hexagon inside it so each vertex touches the circle.

The more sides you add, the more your polygon starts to “hug” the circle, looking less jagged and more round.

Archimedes used clever geometry (no modern trigonometry!) to compute perimeters as he doubled the number of sides step by step.

With 96‑sided polygons, he showed that pi lies between 3 1/7 and 3 10/71, which is about 3.141 and 3.142.

Key facts in bullet points

  • Core shape: regular polygons , not one special curve.
  • Starting point: a regular hexagon in a unit circle.
  • Refinement: doubled sides up to 96‑gons (96‑sided polygons).
  • Method: compare perimeters of:
    • inscribed polygon (too small), and
    • circumscribed polygon (too big).
  • Outcome: tight bounds for pi, accurate to two decimal places.

Meta description:
Discover what shape Archimedes used to approximate pi, how his polygon method worked, and why his 96‑sided construction still fascinates today.
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