Archimedes famously used inscribed and circumscribed polygons to approximate the circumference of a circle. He then repeatedly doubled the numbers of sides to get an approximation for π.

In 1667, James Gregory did the same, but he used areas: He discovered the following beautiful double-recurrence relation that can be used to compute the areas of inscribed and circumscribed *n*-gons:

**Gregory’s Theorem. ***Let I _{k} and C_{k} denote the areas of regular k-gons inscribed in and circumscribed around a given circle. Then for all n, I_{2}_{n} is the geometric mean of I_{n} and C_{n}, and C_{2n} is the harmonic mean of I_{2n} and C_{n}; that is,*

and

We can use these formulas to approximate π. For instance, a square inscribed in a unit circle has area *I*4=2 and a square circumscribed about the unit circle has area *C*_{4}=4. Applying the recurrence relations, we obtain the following sequence of bounds:

n |
In |
Cn |

4 | 2 | 4 |

8 | 2.828427125 | 3.313708499 |

16 | 3.061467459 | 3.182597878 |

32 | 3.121445152 | 3.151724907 |

64 | 3.136548491 | 3.144118385 |

128 | 3.140331157 | 3.14222363 |

256 | 3.141277251 | 3.141750369 |

512 | 3.141513801 | 3.141632081 |

1024 | 3.14157294 | 3.14160251 |

2048 | 3.141587725 | 3.141595118 |

This summer I tweeted this theorem:

Let I_n and C_n be the areas of the inscribed and circumscribed regular n-gons for the unit circle. In 1667James Gr… twitter.com/i/web/status/1…

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Dave Richeson (@divbyzero) June 20, 2018

My friend Tom Edgar—a mathematician at Pacific Lutheran University and a master at finding “proofs without words”—emailed me to see if I wanted to try finding a proof without words of Gregory’s theorem. This is what we came up with.

The two parts of Gregory’s theorem follow from the two parts of the following lemma. We give the proof… without words.

**Lemma. ** and

*Proof.*