Archimedes was born in the Greek colony of Syracuse, Sicily. His father, Phidias, was an astronomer, which likely fostered an early interest in mathematics and astronomy. Syracuse, at the time, was a vibrant center of Greek culture and learning, providing a stimulating environment for the young Archimedes.

Archimedes likely traveled to Alexandria, Egypt, the intellectual capital of the Hellenistic world, to study at the Museum and Library. This institution, founded by Ptolemy I Soter, housed a vast collection of scrolls and attracted brilliant scholars from across the Mediterranean. Here, he would have studied under the successors of Euclid and likely formed lasting connections with other mathematicians and scientists, solidifying his foundations in advanced mathematical concepts.

Archimedes is credited with inventing the Archimedes' screw, a device used for raising water efficiently. This ingenious invention, likely conceived during his time in Egypt or shortly after his return to Syracuse, revolutionized irrigation and drainage systems. Its application extended beyond agriculture, finding use in pumping water out of mines and even as ballast pumps in ships. The Archimedes' screw stands as a testament to his practical engineering skills.

Upon returning to Syracuse, Archimedes dedicated himself to mathematical research. He made groundbreaking contributions to geometry, calculus, and number theory. He derived formulas for the area and volume of various shapes, including the sphere and cylinder. He also developed a method for approximating the value of pi (π) with remarkable accuracy, laying the groundwork for future developments in numerical analysis. His work "On the Sphere and Cylinder" is considered a masterpiece of mathematical reasoning.

The famous anecdote of Archimedes' "Eureka!" moment, while possibly embellished, highlights his genius in problem-solving. According to the story, King Hiero II tasked Archimedes with determining if a crown was made of pure gold or if the goldsmith had mixed it with silver. While pondering this, Archimedes realized that he could determine the volume of an irregularly shaped object by submerging it in water and measuring the displacement. This led to the discovery of the principle of buoyancy, a fundamental concept in fluid mechanics and hydrostatics.

During the Second Punic War, Syracuse allied itself with Carthage and came under siege by the Roman Republic. Archimedes played a crucial role in defending the city with his innovative war machines. These included catapults that hurled massive stones, cranes that could capsize enemy ships, and possibly the legendary 'burning mirrors' that focused sunlight to set Roman ships ablaze. His inventions proved remarkably effective, delaying the Roman conquest of Syracuse for several years.

Syracuse eventually fell to the Romans in 212 BC. Accounts vary regarding the circumstances of Archimedes' death, but the most widely accepted version states that he was killed by a Roman soldier. Plutarch recounts that Archimedes was engrossed in a mathematical diagram when the soldier approached him. Annoyed by Archimedes' disregard, the soldier killed him, despite orders from the Roman general Marcellus to spare his life. The death of Archimedes marked the end of an era, but his legacy continued to inspire generations of scientists and mathematicians.