Archimedes was born in Syracuse, Sicily in 287 B.C. He was the son of Phidias, an astronomer. He was born during the time of Hiero. Archimedes was very fascinated with geometry and spent most of his life finding out things such as the measurement of pi. He was educated at Euclid's school in Alexandria, Egypt. Because of lack of paper he would do his work in dirt and oil, using a stick and his finger to write. He wrote many books on Mathematics, but most of them were lost or destroyed. Archimedes did not hold any public office, however due to his close relationship with King Heiro, King of Sicily, he did donate many of his inventions to the state for war. His inventions seemed to be quite useful in the war. He was unfortunately killed in 212, B.C. in the Second Punic War, by a Roman soldier when the Romans invaded Sicily. He was doing a mathematical problem when a Roman soldier confronted him. He refused to move until his problem was finished so the soldier ran a sword through him. In Archimedes' tomb he requested (before he died) that a sphere containing a cylinder, with the ratio of the two, be inscribed upon it.

Accomplishments
He defined pi as: 3 10/71 < pi < 3 1/7.

Archimedes was not satisfied with the definition of pi as 3 1/7 0r 22/7. This was one of his most important accomplishments. This showed that pi was not equal to 3 1/7. He obtained these numbers by a very difficult process. It also did not help that he did not have the tools that we know have and the knowledge. He did this by circumscribing a regular 96-sided figure (bisecting the sides of a hexagon 4x) inside a circle, and circumscribing a circle inside a 96-sided figure. Archimedes would increase the accuracy of pi by inceasing the number of the sides on the figure. Archimedes increased the number of sides until he couldn't get more precise.
Archimedes discovered Law of Hydrostatics (buoyancy), also known as the Archimedes Principle.

Archimedes discovered buoyancy by a rather interesting story. He did this by helping his friend, King Hiero of Syracuse, trying to figure out the Golden Crown Mystery. The Golden Crown Mystery was a mystery about the king sending a certain amount of gold to the goldsmith for a new crown. When the crown was returned finished, the king thought that not all the gold that was sent to the goldsmith was in his crown. The night of this the king told Archimedes of his problem. When Archimedes went home he discovered what is now known as the Law of Hydrostatics (buoyancy). The Law of Hydrostatics is any object immersed in fluid, is buoyed up by a force equal to the weight of the fluid displayed by the object. Archimedes did this by filling up his tub to the top, and got in. When he got in Archimedes realized that the mass of the water that fell out of the tub was equal that of the mass of his body. The next day he did the same thing with the crown, and the amount of gold supposed to be in the crown. Archimedes found that the gold had more mass than the crown, thus not all the gold was in the crown. An interesting fact about this story is that the goldsmith was beheaded when they found out gold was missing. Aonther interesting fact is when he discovered buoyancy, it is rumored that he ran through the Syracuse streets yelling "eureka", meaning I found it.
Archimedes invented and accomplished many things in his 75 years on the Earth. He invented things such as the Archimedes screw, the catapult, and the burning mirrors.

The Archimedes screw was an invention that allowed people to pump water upwards.

The Arcimedes screw is a machine for pumping water upwards. It was first used to remove water from a hole on a large ship that Archimedes was on.
The first form of the screw invented consisted of a circular pipe inclosing a helix and inclined at an angle of about forty-five degrees horizontally. Its lower end is dipped in the water. The way this works is that the rotation of the device causes the water to rise inside the pipe.
The second form of the Archimedes Screw consists of a helix revolving in a fixed cylinder or a helical tube wound around a shaft. The most modern screws consist of helixes rotating in open inclined troughs. It is now used for pumping sewage in waste-water-treatment plants.
Archimedes discovered the semi-regular shapes (Archimedean Solids).

Archimedes discovered three-dimensional figures that contained either two or more equiangular and equilateral polygons, or the same polygon that met at vertices in different ways. This showed that not all three-dimensional figures do not have all the same polygons and the figures do not all meet in the same way at the vertices. These figures are the: rhombicuboctahedron, icosidodecahedron, truncated dodecahedron, rhombicosidodecahedron, snub dodecahedron,truncated cuboctahedron, truncated icosahedron, snub cube,truncated icosidodecahedron, truncated tetrahedron, trucated octahedron, cuboctahedron, and truncated cube. An example of different polygons making up an Archimedean Solid is the icosidodecahedron. This figure consists of thirty-two bases; they are twenty triangles and twelve pentagons. An example of different kinds of vertices making up an Archimedean Solid is a stella octangula. This figure can be thought of as the union of two Tetrahedra that can be inscribed in a cube. This figure cotains all diagonals of the cube that inscribed the figure faces.

Archimedes also invented the burning mirrors, which were a kind of weapon. It was a hexagonal mirror, which he set similar 4 sided smaller mirrors at interval proportions to the first one. They were connected by hinges. These mirrors made the sun's rays center themselves onto one spot and at any time of year, he could light oncoming ships ablaze with this weapon.
Some other things that Archimedes contributed was that he proved that the volume of a sphere is two-thirds the volume of a circumscribed cylinder. He also had a fascination with levers, and is quoted for saying "Give me a place to stand, and I will move the Earth." By this he meant he could lift the Earth with a lever if there was somewhere he could stand.