10 ## THE AREA OF A CIRCLE## The method of rearrangingTHE AREA
where The equivalent of that formula was known in ancient times. How did they know it? Very likely from the method of rearranging. (See P. Beckmann, Upon moving the shaded triangle to the other side of the parallelogram, it becomes a rectangle with equal base and height. Therefore we now Let us now try to do the same with a circle. We will first cut it up into four equal sectors, and then arrange them as shown above. In the space between each sector, we draw an equal arc. The resulting figure then begins to resemble a parallelogram Its side is The area of that figure is twice the area of the circle. If we now divide the circle into many more sectors, the rearranged figure begins to resemble a parallelogram more closely -- and the side The area of that figure is, again, twice the area of the circle. If we now take an extremely large number of sectors, then the side -- and the figure itself will be indistinguishable from a rectangle whose base is 2π Since the area of that rectangle is twice the area of the circle, then the area of the circle is In the next Topic, we will see how to find the area of a circle by the method of inscribled polygons.
To see the answer, pass your mouse over the colored area.
So much for the formula. Now the question is: How can we find a value for π? Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |