Pythagorean Theorem
The Pythagorean Theorem is basically a formula that is related to the right triangle of Geometry. It goes something like this: If the base line leg of a right triangle is squared, and added to the square of the perpendicular leg___ the total area of these two, will equal the area of the hypotenuse squared. Hence: a² = b² +c² (a = hypotenuse, b = base, & c = perpendicular of any right triangle.) The pictorial geometric figure for this formula is shown as a right triangle, Fig-8, with each side in common with one side of their respective square.
So, I asked myself, what if I replaced the square with a circle or other regular polygon? I found that with the right triangle sides also representing the diameters, or radii of circles, the ratio relationships between the respective circle areas still remain the same as with squares. (Thus it is with semi-circles also.) Fig-9
This is also true for all even sided regular polygons for diameters that are configured in a straight line, radii, sides, and even straight line co-diameters, and co-radii. For odd sided polygons right triangle sides can represent the radii, irregular diameter, and sides since they are straight lines. The Pythagorean relationship of area developed from a right triangle as A = B + C remains the same when using said polygon figures. (Actually there would also be the same relationships between surface areas of the known polyhedrons, and the sphere. More on these, later.)
This means the area for all the regular polygons and the circle when represented for each leg of the right triangle, in similar respective manner, maintain the same ratio to each other as if they were square areas.
Thus it is we add a bit more to our collection of regular polygon and circle similarities. The data presented on this page seems simple and self evident once thought about, and diagramed, but I, have never seen it shown to this extent within any of the math books I have looked at, or own. So, I presented it herein.
