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## #1 2011-08-19 23:57:05

bobbym

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### Squares and circles.

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.

## #2 2011-08-20 01:33:00

anonimnystefy
Real Member

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### Re: Squares and circles.

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 (2011-08-20 01:33:26)

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