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.

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.

Number of doublings

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.

Reviews

Tell us what you think of this calculator.