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Math Help - Equilateral Triangle Inscribed in Circle

  1. #1
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    Equilateral Triangle Inscribed in Circle

    An equilateral triangle is inscribed in a circle of radius r. Express the circumference C of the circle as a function of the length x of a side of the triangle.

    HINT GIVEN:

    First show that r^2 = (x^2/3).
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  2. #2
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    Quote Originally Posted by symmetry View Post
    An equilateral triangle is inscribed in a circle of radius r. Express the circumference C of the circle as a function of the length x of a side of the triangle.

    HINT GIVEN:

    First show that r^2 = (x^2/3).
    Hello,

    I've attached a diagram of the triangle in a circle.

    1. Use Pythagoran theorem:

    h^2+\left( \frac{x}{2}\right)^2=x^2 \Longleftrightarrow h^2=\frac{3}{4} \cdot x^2

    r=\frac{2}{3} \cdot h \Longleftrightarrow h=\frac{3}{2} \cdot r . Thus:

    \frac{9}{4} \cdot r^2= \frac{3}{4} \cdot x^2. Solve for rē and you'll get:

     r^2= \frac{1}{3} \cdot x^2. Therefore: r=\frac{1}{3} \cdot x \cdot \sqrt{3}

    The perimeter of a circle is calculated by:

     c = 2 \cdot \pi \cdot r. Plug in the values you know:

     c = 2 \cdot \pi \cdot \frac{1}{3} \cdot x \cdot \sqrt{3}= \frac{2}{3} \cdot \pi \cdot x \cdot \sqrt{3}


    EB
    Attached Thumbnails Attached Thumbnails Equilateral Triangle Inscribed in Circle-glstg_dreieckinkreis.gif  
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  3. #3
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    ok

    I totally get it.

    Thanks for the picture.

    Can you attach diagrams from now on with your reply? It is a lot easier for me to see what is happening that way.
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