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Math Help - Square inscribed in circle...

  1. #1
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    Square inscribed in circle...

    What is the method for determining the perimeter of a square inscribed in a circle, given the diameter of the circle (4 cm)?
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  2. #2
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by zoso
    What is the method for determining the perimeter of a square inscribed in a circle, given the diameter of the circle (4 cm)?
    The diameter of a circle with perimeter 4 is \frac{4}{\pi} The diameter of the circle is the length from one corner of the square to the opposite corner, in other words it is the hypotenuse of a triangle formed by two sides of the square. The perimeter of the square is the length of a side times 4.. . let's write this out...

    P_{square}=4s

    d^2=s^2+s^2

    d=\frac{c}{\pi}

    therefore...

    d^2=2s^2\Longrightarrow\sqrt{\frac{d^2}{2}=s

    substitute: \sqrt{\left(\frac{c}{\pi}\right)^2\div2}}=s

    \sqrt{\frac{c^2}{\pi^2}\times \frac{1}{2}}=s

    \sqrt{\frac{c^2}{2\pi^2}}=s

    \frac{c}{\pi\sqrt{2}}=s

    therefore

    \frac{4c}{\pi\sqrt{2}}=4s=P_{square}

    rationalize the denominator: \frac{4c\sqrt{2}}{2\pi}=P_{square}

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    Last edited by Quick; July 19th 2006 at 07:07 PM. Reason: adding signature
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  3. #3
    Super Member malaygoel's Avatar
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    The diameter of the circle is the length from one corner of the square to the opposite corner,
    Just for better understanding of the problem(for Zoso),
    Why the above condition has to be true?
    How many squares of different area can be inscribed in the circle? Why?
    How many squares of different perimeter can be inscribed in the circle? Why?

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  4. #4
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by malaygoel
    Just for better understanding of the problem(for Zoso),
    Why the above condition has to be true?
    Right, for Zoso , the above condition is only true for the largest possible square you can inscribe in a circle. If that doesn't help I'll show Zoso a diagram.

    How many squares of different area can be inscribed in the circle? Why?
    How many squares of different perimeter can be inscribed in the circle? Why?

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    Malay
    It's possible to have practically infinite squares inscribed in a circle, maybe if you set up a size restriction I could help you out more?
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  5. #5
    Super Member malaygoel's Avatar
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    It's possible to have practically infinite squares inscribed in a circle, maybe if you set up a size restriction I could help you out more?
    Yes, it is possible have infinite squares, but all of them will have the same area.
    I think are misinterpreting the word 'inscribe'. I means that the vertices of the square lie on the circumference of the circle.
    Right, for Zoso , the above condition is only true for the largest possible square you can inscribe in a circle. If that doesn't help I'll show Zoso a diagram.
    There is no largest square, all of the inscibed squares have the same area.
    The above condition is true because the vertex angle of square is 90 degrees.

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  6. #6
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by malaygoel
    Yes, it is possible have infinite squares, but all of them will have the same area.
    I think are misinterpreting the word 'inscribe'. I means that the vertices of the square lie on the circumference of the circle.
    I said practically infinite.
    There is no largest square, all of the inscibed squares have the same area.
    The above condition is true because the vertex angle of square is 90 degrees.

    Keep Smiling
    Malay
    There is but one possible size of a square per circle that has corners that touch the outside of the circle. I've drawn one below (pay no attention to the fact it's a graph program and looks oval)

    as you can clearly see in the diagram, the diameter of the circle is the distance between two opposite corners of the square.
    Attached Thumbnails Attached Thumbnails Square inscribed in circle...-square-circle.jpg  
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  7. #7
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    The diameter of the circle is the diagnol of the square.
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  8. #8
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    Quote Originally Posted by Quick
    The diameter of a circle with perimeter 4 is \frac{4}{\pi} The diameter of the circle is the length from one corner of the square to the opposite corner, in other words it is the hypotenuse of a triangle formed by two sides of the square. The perimeter of the square is the length of a side times 4.. . let's write this out...

    P_{square}=4s

    d^2=s^2+s^2

    d=\frac{c}{\pi}

    therefore...

    d^2=2s^2\Longrightarrow\sqrt{\frac{d^2}{2}=s

    substitute: \sqrt{\left(\frac{c}{\pi}\right)^2\div2}}=s

    \sqrt{\frac{c^2}{\pi^2}\times \frac{1}{2}}=s

    \sqrt{\frac{c^2}{2\pi^2}}=s

    \frac{c}{\pi\sqrt{2}}=s

    therefore

    \frac{4c}{\pi\sqrt{2}}=4s=P_{square}

    rationalize the denominator: \frac{4c\sqrt{2}}{2\pi}=P_{square}

    Hello, you've discovered my hidden text! hurray for you!
    ~ Q\!u\!i\!c\!k

    so if the diameter of the circle IS 4 cm, the result would be  16\sqrt {2} cm or something near 22 cms for the perimeter?
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  9. #9
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by zoso
    so if the diameter of the circle IS 4 cm, the result would be  16\sqrt {2} cm or something near 22 cms for the perimeter?
    It would be...

    \frac{4c\sqrt{2}}{2\pi}=\frac{4(4)\sqrt{2}}{2\pi}=  \frac{16\sqrt{2}}{2\pi}=\frac{8\sqrt{2}}{\pi}<br />
\approx<br />
3.601265265cm
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  10. #10
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    k, I see what I did wrong.
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  11. #11
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    The diameter is 4cm...not the circumference. So, using pythagoras' theorem, 2 squard+ 2 sqard=8
    root 8= 2.828
    2.828 x 4= ~11.3
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  12. #12
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by thedarktemplar
    The diameter is 4cm...not the circumference.
    It's a good thing you noticed. This actually makes the equation easier...
    Quote Originally Posted by Quick
    The diameter of a circle with perimeter 4 is \frac{4}{\pi} The diameter of the circle is the length from one corner of the square to the opposite corner, in other words it is the hypotenuse of a triangle formed by two sides of the square. The perimeter of the square is the length of a side times 4.. . let's write this out...

    P_{square}=4s

    d^2=s^2+s^2

    d=\frac{c}{\pi}

    therefore...

    d^2=2s^2\Longrightarrow\sqrt{\frac{d^2}{2}}=s
    expand on that and you'll get...
    \frac{4d}{\sqrt{2}}=4s=P_{square}
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