Archimedes was born about 287 BC, so he was a little younger than Euclid. His father was an astronomer. Archimedes was related to the tyrant who ruled Syracuse, on the island of Sicily, and he lived his whole life in Sicily, except when he went to study at the University of Alexandria in Egypt, where he probably met and worked with the other great scientists of his time: Euclid, Aristarchus, and Eratosthenes. Archimedes worked mainly in a losing battle to defend his city-state from the Romans, who were attacking Syracuse in the course of the First Punic War.
Archimedes invented, or people said he had invented, a bunch of different kinds of machines. The most important of these machines was the screw pump, which uses a screw to lift water from one place to another.
But Archimedes was also interested in why things worked, and whether they would work the same way every time. He explained why levers worked, and he worked on getting a more accurate number for pi. The accomplishment Archimedes himself was most proud of was that he proved that if you fit a sphere inside a cylinder, the sphere will have two-thirds the volume and two-thirds of the surface area of the cylinder.
Archimedes died when the Romans were conquering Syracuse; a Roman soldier killed him.
To find out more about Archimedes, check out these books from Amazon.com or from your library:
Greek and Roman Science, by Don Nardo (1998). Nardo has written a lot of good books about the ancient world for kids; this one is no exception.
Ancient Science: 40 Time-Traveling, World-Exploring, History-Making Activities for Kids, by Jim Wiese (2003). Activities, as the title says - how to make your own sundial, and so on. The author is a science teacher.
Early Greek Science: Thales to Aristotle, by Geoffrey Lloyd (1974).
History of Greek Mathematics: From Aristarchus to Diophantus, by Thomas L. Heath (1921, reprinted 1981). A lot of Euclid, but also describes who the other major Greek mathematicians were and what they did.
Episodes from the Early History of Mathematics, by Asger Aaboe (1997).