Pi Polygon Method Calculator (Archimedes)
Approximate pi by bounding it between inscribed and circumscribed regular polygons of a unit circle. Watch Archimedes method converge as the side count doubles, step by step.
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Input
Approximate pi by squeezing it between the perimeters of regular polygons inscribed in and circumscribed about a unit circle. Enter how many times to double the sides.
The start is a hexagon (6 sides). Each doubling multiplies the side count by two. Allowed range is 0 to 25.
Result
Pi approximation from a 96-sided polygon
3.141873275268
Inscribed (lower bound)
3.141031950891
Circumscribed (upper bound)
3.142714599645
Error
0.000280621678
Pi lies between the inscribed value 3.141031950891 and the circumscribed value 3.142714599645, a bracket width of 0.001682648755.
Convergence by doubling
As the side count doubles, the inscribed and circumscribed values approach each other and the error shrinks.
| Sides | Inscribed | Circumscribed | Error |
|---|---|---|---|
| 6 | 3 | 3.464101615138 | 0.090458153979 |
| 12 | 3.10582854123 | 3.215390309173 | 0.019016771612 |
| 24 | 3.132628613281 | 3.159659942098 | 0.0045516241 |
| 48 | 3.139350203047 | 3.146086215131 | 0.001125555499 |
| 96 | 3.141031950891 | 3.142714599645 | 0.000280621678 |
Inscribed value equals n times sin(180 degrees over n) and circumscribed value equals n times tan(180 degrees over n). The approximation is their average, compared with the true value 3.14159265359.
How it works
- The inscribed regular n-gon half-perimeter is n times sin(180 degrees over n) and the circumscribed half-perimeter is n times tan(180 degrees over n), giving a lower and upper bound on pi.
- Starting from a hexagon and doubling the sides (6, 12, 24, 48 and so on), the polygons hug the circle more tightly, so the gap between inscribed and circumscribed values shrinks toward pi.
- The headline approximation is the average of the inscribed and circumscribed values, and the error is its absolute difference from the true value of pi.
- Archimedes of ancient Greece used a 96-sided polygon to prove that pi lies between roughly 3.1408 and 3.1429.
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