Archimedes' Mathematics
Euclid's geometry (fl. c. 300 BC)
The circumference of a circle is pi times the circle's diameter (definition of pi). The value of pi was known to be approximately 3. Until Archimedes arrived, no one had attempted to calculate a more accurate value.
The area of a circle is a constant (?) times the square of the circle's radius.
The volume of a cylinder is the area of the circular base times its height (due to Eudoxus?). The volume of a cone is 1/3 of the volume of the cylinder that surrounds it (due to Eudoxus). Archimedes Traps Pi
Pi is defined as the ratio of the circumference of a circles to its diameter.
(pi as a symbol wasn't used until 1706)
How did Archimedes estimate the value of pi?
He drew two polygons around the circle's center - one outside the circle (circumscribed) so its perimeter was greater than the circle's, and one inside the circle (inscribed) so its perimeter was less than the circle's.
Fig: http://physics.weber.edu/carroll/archimedes/pi.htm
Using a similar method, Archimedes proved that the area of a circle of diameter D is equal to the area of a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.
Fig: http://physics.weber.edu/carroll/archimedes/images/circlearea.gif