Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Squares and circles.

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!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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 (2011-08-19 03:33:26)

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