Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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.

Thanks,

Corinna


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
 Check out our web site!  http://mathforum.org/dr.math/   
    
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

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/