Fraction of the Original Square

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Kaycee

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Aug 26, 2005
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start with a square piece of paper. draw the largest circle possible inside
the square, cut it out and discard the trimmings. draw the largest square
possible inside the circle, cut the square out and discard the trimmings.
what fraction of the original square piece of paper has been cut off and
thrown away?

Can Someone Please Help Me out Please
Thank You
 
Make the original square 1 by 1; so area = 1 ; ok?

The circle inside will have diameter = 1 ; ok?

So the 2nd square will have diagonals = 1 ; ok?

So find area of a square with diagonals = 1
 
circleandsquare1qk.gif


The square inside the circle will have diagonals of 1 unit.

Use Pythagoras, what squared equals 1/2?. That'll be a side length of the

square. Use that in the theorem.
 
Square Area: S*S
Inside Circle: d = s, so r = s/2
So, A of circle: \(\displaystyle \pi\frac{s^2}{4}\)

I don;t know your level of math, but:

The square touches at \(\displaystyle \theta = \frac{\pi}{4}\),\(\displaystyle \frac{3\pi}{4}\),\(\displaystyle \frac{5\pi}{4}\), and \(\displaystyle \frac{7\pi}{4}\)

The x and y components are fnctions of sine and cosine of the given \(\displaystyle \theta\)'s.

The points of the square are thus at:
given the radius of the circle is \(\displaystyle \frac{s}{2}\)

Q1: \(\displaystyle (\frac{s\sqrt{2}}{4},\frac{s\sqrt{2}}{4})\)
Q2: \(\displaystyle (-\frac{s\sqrt{2}}{4},\frac{s\sqrt{2}}{4})\)
Q3: \(\displaystyle (-\frac{s\sqrt{2}}{4},-\frac{s\sqrt{2}}{4})\)
Q4: \(\displaystyle (\frac{s\sqrt{2}}{4},-\frac{s\sqrt{2}}{4})\)

The distance between any adjacent corner to another is the length of the side of the new square, so use the distance formula. I will use Q1 and Q2.

S = \(\displaystyle \sqrt{(\frac{s\sqrt{2}}{4}+\frac{s\sqrt{2}}{4})^2 + (\frac{s\sqrt{2}}{4}-\frac{s\sqrt{2}}{4})^2}\)

Now, the area of the new square is:

S*S = \(\displaystyle (\frac{s\sqrt{2}}{4}+\frac{s\sqrt{2}}{4})^2 + (\frac{s\sqrt{2}}{4}-\frac{s\sqrt{2}}{4})^2 = (\frac{s\sqrt{2}}{4}+\frac{s\sqrt{2}}{4})^2 = (\frac{s\sqrt{2}}{2})^2 = \frac{s^2}{2}\)

Now what is the ratio of this area to the original area?
 
Another hint: Take that first picture galactus drew, and draw the second square inside the circle, but rotated 45°. That is, draw the second square so its corners are pointing in the four compass directions.

Now ignore the "circle" discussion, and just look at the two squares. Can you see how much of the original square is lost when you cut around the inner square?

Eliz.
 
Hello, Kaycee!

This is what Stapel suggested . . .
Code: 
      +-------*---*---*-------+
      |   *     /   \     *   |
      |       /       \       |
      |*    /           \    *|
      |   /               \   |
      * /                   \ * 
      *                       *
      * \                   / *
      |   \               /   |
      |*    \           /   * |
      |       \       /       |  
      |   *     \   /     *   |
      +-------*---*---*-------+
Nice one, Eliz !
 
soroban, what do you use to draw such nice pictures? Do you really bother to type it all out?
 
Hello, daon!

I do type it out, but I've invented my own "grid" systems which helps a bit.

After typing [ code], I fill the next few lines with spaced periods:

. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .


Then I start typing my diagram, using the "graph paper" I created.

. . . . A . . . . .
. . . . * . . . . .
. . . ./.\. . . . .
. . . / . \ . . . .
. . ./. . .\. . . .
. . * - - - * . . .
. . B . . . C . . .

I click on "Preview" to see what it looks like:
Code: 
. . . . A . . . . .
. . . . * . . . . .
. . . ./.\. . . . .
. . . / . \ . . . .
. . ./. . .\. . . .
. . * - - - * . . .
. . B . . . C . . .

If I'm satisfied with the result, I "erase" the periods.

. . . . A
. . . . *
. . . ./ \
. . . / . \
. . ./. . .\
. . * - - - *
. . B . . . C
Code: 
        A
        *
       / \
      /   \
     /     \
    * - - - *
    B       C
Yes, it's more work than most people care to invest,
. . but I enjoy the challenge.
 
Thats great! I always see your nice graphs and wondered what software you used to do them, ha. Thanks.
 
Code: 
.........*..............
......*.....*............
....*.........*........
....*.........*.......
......*.....*...........
.........*.............
Beginner's luck :roll:
 
Not bad, Denis . . . not bad at all!

I like to "flatten" the extremes of my circles.

. . . * * * . .
. . .*. . . .*. . .
. .*. . . . . .*.
. * . . . . . . *
. * . . .o. . . *
. * . . . . . . *
. .*. . . . . .*.
. . .*. . . .*. .
. . . .* * * . .

Code: 
       * * *
     *       *
   *           *
  *             * 
  *      o      *
  *             * 
   *           *
     *       *
       * * *
 
just about perfect, Soroban; looks like pi = an even 3 :shock:
 
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