Area of a Circle - The Polygon Method
There are several methods to derive the formula for the area of a circle. Here's the polygon method..
What's a polygon?
Take a sheet of paper and place the tip of your pencil anywhere on the sheet. Now draw a straight line. It can be any length and in any direction, but it must be straight, curves are not allowed. Now repeat without lifting your pencil from the sheet continue to draw straight lines, and don't let any lines touch or cross. So you have a "chain" of straight lines all connected to each other.
To make this chain into a polygon you only need do one more thing. Make sure the last line you draw finishes at the exact point you started. So the chain is closed.
Closing the chain makes this a polygon and produces one of its critical properties.. it has an inside and an outside, so therefore it has an area.
A line of a polygon is called a "face" or "side" or "edge" and the point where two edges meet is called a "vertex".
Of course, calculating the area of a general polygon is insanely difficult, so we're not even going to try. Instead we'll calculate the area of a special class of polygons, these are the "regular convex" polygons.
"Regular" means all the sides are of equal length and the angles between them are all equal.
"Convex" means that the polygon "bulges out", or to be more precise.. if you pick any two points inside the polygon and draw a line between them it will not cross any of the sides. An example will help..
A square is a regular convex polygon with 4 faces.
So, what about the area of a square? If it has side d we know that the area is given by the famous formula..
But let's write this differently. Does the square have a "circumference" (that is, the total length of its boundary)? Yes, it's just 4*d. Does the square have a "radius"? We'll define the radius as the length of a line from the center of the polygon to the center of a face. For a regular convex polygon this distance is the same for all faces so it's a well defined number. The diameter is just twice the radius. So the radius of the square is d/2. Now we notice that..
So, for a square we've produced a simple formula for the area. A square with circumference c and radius r has..
This seems like a neat formula. It's elegant and simple. Could it be true for other polygons?
Yes. This is true for any regular convex polygon with n sides!
So we've just got the area of any regular convex polygon, no matter how many sides it has!
But it gets better. As the number of sides increases the polygon begins to look more and more like a circle. It's not a circle of course, but as the number of sides n approaches infinity the polygon approaches a circle. When n gets very big our area formula is still valid so the area of a circle is..
But by definition pi=c/(2*r) or c=2*pi*r and so we get the famous formula for the area of a circle..
area of a circle=2*pi*r*r/2=pi*r^2
Which is the correct formula for the area of a circle.
Content written and posted by Ken Abbott firstname.lastname@example.org