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Archimedes' Method of Estimating Pi

Date: 5/29/96 at 21:24:35
From: Larry Sherman
Subject: Archimedes' method of estimating pi - ?

Tell me about Archimedes' method for estimating pi using inscribed and 
circumscribed polygons about a circle.



Date: 5/30/96 at 14:38:30
From: Doctor Darrin
Subject: Re: Archimedes' method of estimating pi - ?

Archimedes knew that the area of a circle was pi * r^2.  He estimated 
the value of pi by estimating the area of a circle with radius 1 (and 
area pi). 

To do this, he would calculate the area of a regular polygon inscribed 
in the circle. Since the polygon would be entirely contained in the 
circle, it would have an area less than the area of the circle. For 
instance, if we inscribed a regular hexagon in a circle of radius 1, 
we could divide the hexagon into 6 equilateral triangles, each having 
sides of length one. The area of the triangles would then be about 
.433, so the area of the hexagon is 6*.433=2.60.  Thus, we see that pi 
is greater than 2.6.  

If we circumscribe a hexagon around a circle, then we can divide it 
into six equilateral triangles each having area .577, so the hexagon 
has area 3.46. Since the circle is inside the hexagon, it has area 
less than 3.46, so we see that pi is less than 3.46.  

Archimedes did much better than this - he used regular polygons with 
96 sides, and found that pi is between 3+(10/71) and 3+(1/7).  

I hope this helps.

-Doctor Darrin,  The Math Forum
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Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School History/Biography
Middle School Conic Sections/Circles
Middle School Geometry
Middle School History/Biography
Middle School Pi

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