1. Archimedes The area of the unit circle

2.  Archimedes (287-212 B.C.) sought a way to compute the area of the unit circle. He got the answer correct to within less than 1/10 of 1%.

3.  First, he noted that the area of the circle was greater than the area of any polygon inscribed inside it. In particular, it is greater than the area of an inscribed hexagon.

4.  The area of the hexagon, is exactly six times the area of an equilateral triangle of side 1. 1

5.  The area of 1 triangle =  The area of the hexagon =  The area of 1 triangle =  The area of the hexagon =

6.  In a similar manner, the area of the circle is less than the area of the circumscribed hexagon, which is  So, we have the area of the circle,, must be between 2.598 and 3.464.  In a similar manner, the area of the circle is less than the area of the circumscribed hexagon, which is  So, we have the area of the circle,, must be between 2.598 and 3.464.

7.  To get more precise estimates, Archimedes used polygons with more sides. With dodecagons (12-sided polygons) we get  To get more precise estimates, Archimedes used polygons with more sides. With dodecagons (12-sided polygons) we get

8. # sidesArea of inscribed polygon Area of circumscribed polygon 62.5980763.464102 123.0000003.215390 243.1058293.159660 483.1326293.146086 963.1393503.142715 1803.1409553.141912 3603.1414333.141672 7203.1415533.141613 Archimedes stopped here 