Archimedes' Method of Estimating PiDate: 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/ |
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