Why Pi?Date: 01/25/2001 at 11:41:06 From: crystal Subject: Why Pi? Dr. Math, I was just wondering... Why do we use pi when we calculate the circumference and area of a circle? I think one of my professors once told my class but I can't remember and am curious. Thanks. Date: 01/25/2001 at 13:06:19 From: Doctor Peterson Subject: Re: Why Pi? Hi, Crystal. There are two questions here, with very different answers. First, for the circumference, it's because we DEFINE pi as C/D, so we can write C = pi D automatically. There's a trick hidden behind that definition, though: how do we know that C/D is the same for every circle? That takes a bit of proof, and leads to some interesting ideas; look in the Dr. Math FAQ on pi: Pi = 3.14159... http://mathforum.org/dr.math/faq/faq.pi.html or in the following answers in particular, for an explanation: Why is Pi a Constant? http://mathforum.org/library/drmath/view/57828.html Einstein, Curved Space, and Pi http://mathforum.org/library/drmath/view/55198.html Is Pi a Constant in Non-Euclidean Geometry? http://mathforum.org/library/drmath/view/55021.html For the area, there's a nice way to see why the formula should be what it is. Let's think about regular polygons first, and look at the relation between their areas and perimeters. Any n-sided polygon can be broken into n isosceles triangles like this: +-----+ / \ / \ / \ / \ +-----+-----+ + \ / \ / /|\ \ / \ / / |a\ +-----+ +--+--+ s Each of these triangles has a base that is equal to a side s of the polygon, and a height a (called the apothem); the total area is A = n * sa/2 This can be rearranged as A = (ns)a/2 and since ns is just the perimeter P of the polygon, this means A = Pa/2 Now make n very large, and a will be very close to the radius r of the circle the polygon is becoming. We can see (and could prove more carefully if we took the time) that for a circle, A = Cr/2 where C is the circumference (perimeter) and r is the radius. But since we know C = 2 pi r this becomes A = 2 pi r * r/2 = pi r^2 We're done! Because we could find the area of a polygon using its perimeter, we can find the area of a circle using its circumference, and that uses pi. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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