Problem was just posed by ganesh.
Find the ratio of the areas of the incircle and circumcircle of a square.
I know there is a lot of ways to do this but supposing you did not have any idea how to solve this. Geogebra to the rescue!
1) Make a 4 sided regular polygon. ( a square )
2) Use the 3 point circle option to draw the outer circle using 3 of the vertices of the square.
3) Draw the diagonal line segments to get the center of the square.
4) Put a point F on the square. Make that line segment from the center to F parallel with the x axis.
5) use the point and radius circle option to make a point from the center of the square to f.
6) Get areas of both circles.
7) Take the ratio:
8) use one of the free vertices to expand the inner circle. Find the new areas. What do you deduce?
Looks like the ratio of the areas is 1 / 2. Not rigorous but definitely enough to go to war with!
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
i solved this before,and it is very easy.
the radius of the inner circle is a/2 where a is the side of the square.so it's area is A1=pi*a^2/4
the radius of the outer circle is a/sqrt(2).it's area is A2=pi*a^2/2.
Last edited by anonimnystefy (2011-08-19 03:33:26)
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