Archimedes The area of the unit circle.  Archimedes (287-212 B.C.) sought a way to compute the area of the unit circle. He got the answer correct to.

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Presentation transcript:

Archimedes The area of the unit circle

 Archimedes ( 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%.

 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.

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

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

 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 and  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 and

 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

# sidesArea of inscribed polygon Area of circumscribed polygon Archimedes stopped here 