Short answers, pulled from the story.
Archimedes of Syracuse was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the city of Syracuse in Sicily. He is widely regarded as one of the leading scientists of classical antiquity and one of the greatest mathematicians of all time. He was born around 287 BC and died in 212 BC.
Archimedes approximated pi in Measurement of a Circle by drawing a regular hexagon outside a circle and another inside it, then progressively doubling the number of sides. After four steps, when each polygon had 96 sides, he determined that pi lay between roughly 3.1408 and 3.1429, consistent with its actual value near 3.1416.
Eureka means I have found it. According to Vitruvius, Archimedes shouted it after stepping into a bath and noticing the water rose higher the deeper he sank, realizing he could use displacement to measure the volume of a golden wreath suspected of containing silver. He was so excited that he ran into the streets without dressing.
Archimedes died during the sack of Syracuse in 212 BC, killed by a Roman soldier despite orders from Marcus Claudius Marcellus that he should not be harmed. The oldest account, from Livy, says a soldier who did not recognize him struck him down while he was drawing figures in the dust.
The Archimedes Palimpsest is a 174-page 13th-century parchment of prayers written over erased 10th-century copies of treatises by Archimedes. Identified by Johan Ludvig Heiberg in 1906, it holds seven works, including the only surviving Greek copy of On Floating Bodies and the sole source of The Method of Mechanical Theorems. It sold at auction on the 29th of October 1998 for 2.2 million dollars.
Archimedes valued most the relationship between a sphere and a circumscribed cylinder of the same height and diameter, where the sphere's volume and surface area are two-thirds those of the cylinder including its bases. He asked that a sphere and a cylinder be placed on his tomb, which Cicero later found near the Agrigentine gate in Syracuse.
In The Sand Reckoner, Archimedes calculated a number greater than the grains of sand needed to fill the universe, concluding it was 8 vigintillion. To do so he devised a counting system based on powers of a myriad of myriads, equal to 100 million, demonstrating that mathematics could represent arbitrarily large numbers.