Archimedes determined the area of a circle while using the Pythagorean Theorem to find the areas of 2 regular polygons; The polygon wrote within the circle and the polygon within where the circle was defined. Since the real area of the circle is in between the areas of the writin and defined polygons, the areas of the polygon gave higher and lower bounds for the area of the circle. Archimedes knew that he had not yet found the value of pi but simularity within those limits. Archimedes showed that pi is between 3 1/7 and 3 10/71.
King Hiero had instructed a new royal crown for which he provided solid gold to the goldsmith. When the crown arrived, King Hiero was suspicious that the goldsmith only used a fraction of the gold, kept the rest for himself and added silver to make the crown the correct weight. Archimedes was asked to determine whether or not the crown was pure gold without causing damage to it in the process. Archimedes was uncertain but found inspiration while taking a bath. He noticed that the full bath overflowed when he lowered himself into the bathtub, and suddenly realized that he could measure the crown's volume by the amount of water it made difference. He knew that since he could measure the crown's volume, all he had to do was discover its weight in order to calculate its density and so find the its purity. Archimedes was so excited about his discovery that he ran down the streets of Syracuse naked shouting, “Eureka!” which meant “I've found it!” in Greek.
Volume=pi radious2 h would be the formula.