Polygons Explained in 5 Minutes
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. Keep going, and don't let the lines touch or cross. So you have a "chain" of straight lines all connected to each other.
To make this 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 straight line between them it will not cross any of the sides. An example will help..
A square is a regular convex polygon with 4 sides. A "stop sign" is a regular convex polygon with 8 sides.
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 "perimeter"? Yes, it's 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 perimeter 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 found the area of any regular convex polygon, no matter how many sides it has!
But it gets even 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. Is our area formula still valid as n goes to infinity?
Yes! The area of a circle is c*r/2 where c is the circumference, which is just the "perimeter" of a circle.
So the area of any regular convex polygon with n sides is c*r/2 even when n is infinite and the polygon becomes a circle.
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Content written and posted by Ken Abbott email@example.com