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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,442

Problem was just posed by ganesh.

ganesh wrote:

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.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,327

hi bobbym

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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