Formula for the Area of a Circle

Date: 12/17/98 at 23:37:57
From: Kismet
Subject: How do you get the area of a circle?

I haven't figured any of it out, but I want to know how to do it. 
Please help.

Date: 12/18/98 at 12:03:44
From: Doctor Peterson
Subject: Re: How do you get the area of a circle?

Hi, Kismet. I'm not sure whether you're asking for the formula for the 
area of a circle, or for an explanation of how it works. I'll give you 
both.

The formula is very simple:

    A = pi * r^2

which means the area is Pi (3.14159...) times the square of the radius. 
In a book it would look more like this:

        __  2
    A = || r

To use this formula, just measure the radius of the circle (which is 
half the diameter), square it (multiply it by itself), and then 
multiply the result by 3.14.

There's an interesting way to see why this is true, which may help you 
remember it. (Though the easiest way to remember the formula is the 
old joke: "Why do they say "pie are square" when pies are round?")

Picture a circle as a slice of lemon with lots of sections (I'll only 
show 6 sections, but you should imagine as many as possible):

         *       *
      *   \     /   *
     *     \   /     *
    *       \ /       *
    *--------+--------*
    *       / \       *
     *     /   \     *
      *   /     \   *
         *       *

Now cut it along a radius and unroll it:

        /\      /\      /\      /\      /\      /\
       /  \    /  \    /  \    /  \    /  \    /  \
      /    \  /    \  /    \  /    \  /    \  /    \
     /      \/      \/      \/      \/      \/      \
    **************************************************

All those sections (technically called sectors of the circle) are close 
enough to triangles (if you make enough of them) that we can use the 
triangle formula to figure out their area; all together they are

    A = 1/2 b * h = 1/2 C * r

since the total base length is C, the circumference of the circle, and 
the height of all the triangles is r, the radius (if the triangles are 
thin enough). You should know that the circumference is pi times the 
diameter, or

    C = 2 * pi * r

(this is actually the definition of pi), so the area is just

    A = 1/2 (2 * pi * r) * r = pi * r^2

In other words, the area of a circle is just the area of a triangle 
whose base is the circumference of the circle, and whose height is 
the radius of the circle.

What I've just done gets pretty close to algebra, which you haven't 
learned yet, but if you think about it (and maybe try actually 
measuring some real circles, or even make some lemonade) you should 
be able to see what I mean.
 
You probably didn't know that the area of a circle is the same as the 
area of a triangle!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/