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Thread: Area of Square in Circle

  1. #1
    Senior Member harpazo's Avatar
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    Area of Square in Circle

    There is a shaded square inside a circle represented by the equation x^2 + y^2 = 9. Find the area of the square in the circle.

    I seek the first-two steps.
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    MHF Contributor MarkFL's Avatar
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    Re: Area of Square in Circle

    I am assuming the square is inscribed within the circle, and so the diagonal of the square is equal to the diameter of the circle.

    Let $\displaystyle D=2r$ by the diameter of the circle. The area $\displaystyle A$ of the square is:

    $\displaystyle A=s^2$

    And we know:

    $\displaystyle 2r=\sqrt{2}s\implies s=\sqrt{2}r\implies A=2r^2$

    Can you proceed?
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    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by MarkFL View Post
    I am assuming the square is inscribed within the circle, and so the diagonal of the square is equal to the diameter of the circle.

    Let $\displaystyle D=2r$ by the diameter of the circle. The area $\displaystyle A$ of the square is:

    $\displaystyle A=s^2$

    And we know:

    $\displaystyle 2r=\sqrt{2}s\implies s=\sqrt{2}r\implies A=2r^2$

    Can you proceed?
    A = 2r^2

    I know that x^2 + y^2 = r^2.

    So, r^2 = 9 leading to an r value of 3.

    A = 2r^2

    A = 2(3)^2

    A = 2(9)

    A = 18 units^2
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    MHF Contributor MarkFL's Avatar
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    Re: Area of Square in Circle

    What would the area of a square be that circumscribes a circle of radius $\displaystyle r$? (This is there the circle is inside the square)
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    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by MarkFL View Post
    What would the area of a square be that circumscribes a circle of radius $\displaystyle r$? (This is there the circle is inside the square)
    If I know r, then A = 2r^2 yields the area of a square inside a circle.
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    Re: Area of Square in Circle

    Quote Originally Posted by harpazo View Post
    If I know r, then A = 2r^2 yields the area of a square inside a circle.
    Rethink that! Are you missing some brackets?
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    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by Debsta View Post
    Rethink that! Are you missing some brackets?
    A = 2(r)^2
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    MHF Contributor MarkFL's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by harpazo View Post
    If I know r, then A = 2r^2 yields the area of a square inside a circle.
    If the circle is inscribed within the square, then the side length of the square will be equal to the diameter of the circle, thus:

    $\displaystyle A=(2r)^2=4r^2$

    Now, given the result for the original problem you posted, we now know:

    $\displaystyle 2r^2<\pi r^2<4r^2$

    Or:

    $\displaystyle 2<\pi<4$

    How could we improve these bounds on $\displaystyle \pi$?
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    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by MarkFL View Post
    If the circle is inscribed within the square, then the side length of the square will be equal to the diameter of the circle, thus:

    $\displaystyle A=(2r)^2=4r^2$

    Now, given the result for the original problem you posted, we now know:

    $\displaystyle 2r^2<\pi r^2<4r^2$

    Or:

    $\displaystyle 2<\pi<4$

    How could we improve these bounds on $\displaystyle \pi$?
    Sorry but I don't follow? Is my answer to the original question right?
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    MHF Contributor MarkFL's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by harpazo View Post
    Sorry but I don't follow? Is my answer to the original question right?
    Yes, you answered the original problem correctly, and is why I thanked your post.

    Consider the following diagram:



    Can you see that the side length of the larger square is equal to the diameter of the circle inscribed within?
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    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by MarkFL View Post
    Yes, you answered the original problem correctly, and is why I thanked your post.

    Consider the following diagram:



    Can you see that the side length of the larger square is equal to the diameter of the circle inscribed within?
    Yes, I can what you are saying here.
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  12. #12
    MHF Contributor MarkFL's Avatar
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    Re: Area of Square in Circle

    Do you see how the rest of what I posted in post #8 then follows?
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    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by MarkFL View Post
    Do you see how the rest of what I posted in post #8 then follows?
    Yes.
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    MHF Contributor MarkFL's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by harpazo View Post
    Yes.
    Okay, then can you think of a way we could improve our bounds on the value of pi?
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  15. #15
    Senior Member harpazo's Avatar
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    Re: Area of Square in Circle

    Quote Originally Posted by MarkFL View Post
    Okay, then can you think of a way we could improve our bounds on the value of pi?
    Improve our bounds???
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