Curve :

Archimedean spiral

It has many uses. It can be intriguing.

Proof :

Lune of Hippocrates, named after Hippocrates of Chios :

Hippocrates of Chios wanted to solve the "quadrature" of the circle but solved the lune instead.

Euclid, Aristotle and many other mathematicians liked the proof a lot.

Formula :

Eddington number.

The Sand Reckoner (Ψαμμίτης / Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers. The work, also known in Latin as Archimedis Syracusani Arenarius & Dimensio Circuli, which is about 8 pages long in translation, is addressed to the Syracusan king Gelo II (son of Hiero II), and is probably the most accessible work of Archimedes; in some sense, it is the first research-expository paper.[1]

Arhimedes and Eddington were both right ! :)

As proof, I also like the one measuring the craters in the moon.

Galileo (

http://www.compadre.org/osp/items/detail.cfm?ID=11196).