Square Within Circle With No Numbers Given

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geekily

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This problem is kind of the opposite of the one I saw was posted recently, but try as I might, I'm still unable to figure it out after reading that post. "Imagine the largest square plug that fits into a circular hole. How well does the plug fit? That is, what percentage of the circular hole does the square plug occupy?"

I know I need area of the circle - area of square = amount not occupied (x) so I can do x/360 to get the % not occupied (y) and then 100 - y to get % occupied. I plugged in my formulas, pie r squared - base(height). However, since I have no numbers to work with, I don't know where to go next. I tried just arbitrarily assigning some numbers, but that got me even more confused. If anyone could point me in the right direction, I'd really appreciate it. Thanks!
 
For any max square within a circle, the circle's diameter is going to be found using the Pythagorean theorem: a^2 + b^2 = c^2, where a = L, b = W and c = circle's diameter.

\(\displaystyle \L \sqrt{L^2\,+\,W^2}\) where L and W are the length and Width of the square. This simplifies to \(\displaystyle \L \sqrt{2L^2} = \sqrt{2}\sqrt{L^2} = L\sqrt{2}\)

Because L = W for a square.

Use this information to solve for the dimensions of your square or circle:

Use the formula (pi)(r)^2 to get your area of the circle, and L*W to get the area of your square. When you have a_1 and a_2, than

(what % of area 1 = area 2)?

x(a_1) = (a_2)
x(100) = ((a_2)/(a_1))100 = percentage of area 1 that is being taken up.

Right? Correct me if I'm wrong :!:
 
Yes, but how do I get my percentage or anything for that matter if I don't have any numbers to work with?
 
Like I have shown.... I wasn't given any numbers, was I?

If you don't have values, than you can only use variables.
 
Hello, geekily!

Imagine the largest square plug that fits into a circular hole.
How well does the plug fit?
That is, what percentage of the circular hole does the square plug occupy?
Click to expand...

Draw a circle with center \(\displaystyle O.\)
Inscribe a square \(\displaystyle ABCD.\)
Draw diagonal \(\displaystyle AOC.\)
Let \(\displaystyle r\) = the radius. \(\displaystyle \;OA\,=\,OC\,=\,r\)

Let \(\displaystyle x \,=\,AB\,=\,BC\), the side of the square.

In right triangle \(\displaystyle ABC\), we have: \(\displaystyle \:x^2\,+\,x^2\:=\:(2r)^2\)

. . \(\displaystyle 2x^2\:=\:4r^2\;\;\Rightarrow\;\;x^2\:=\:2r^2\;\;\Rightarrow\;\;x\:=\:r\sqrt{2}\)


The area of the square is: \(\displaystyle \:x^2\:=\:(r\sqrt{2})^2\:=\:2r^2\)

The area of the circle is: \(\displaystyle \:\pi r^2\)

The ratio is: \(\displaystyle \L\:\frac{2r^2}{\pi r^2} \:=\:\frac{2}{\pi}\:=\:0.636619772 \:\approx\:63.7\%\)

 
Thank you both so much, you were both very helpful. Soroban, I think it really helped to be able to see it written out like that, and jwpaine, you were very helpful, as well.

Now my next problem is to turn it around - circle inside the square - so I'm trying to turn around what you taught me to figure it out: I drew a diagram, labeled the square ABCD, and the middle of the circle O. However, I can't draw diagonal AOE because there are gaps between the circle and the right angle. So instead, I put point E between points A and B and point F between points C and D, and then drew EOF. That doesn't seem to do me any good, though, because it doesn't give me the right triangle I need to do Pythagorean Theorem. Where would I go from here?

Again, thank you both so much!
 
you don't need the Pythagorean theorem for this problem.

look again at your sketch ... the diameter of the circle is the same length as the side of the square.
 
Oooh, so I can just make a ratio again? Thank you so much - I'm starting to feel so much better about this stuff.

...Wait, I set my ratio up as pie r squared / 2 r squared, and then canceled out the r squared to end up with pie / 2. However, that came out to about 157%. I went back to see what I did wrong, and I thought maybe it had to be pie r squared / (2r)squared, so I distributed the square before I canceled it out and got pie / 4. That gave me about 78.5%. Is that right?

Thanks so much!
 
correct.

btw ... pi is the transcendental number that is the ratio between the circumference and the diameter of a circle.

pie is something you eat. Banana Cream for me, please.
 
*facepalm*

Oh, wow. I knew that, I swear. I'm so ashamed, being the grammar freak that I claim to be and all.

Thanks for your help. :)
 
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