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Thread: If the radius of the small circle is 8 cm, what is the circumference of the...

  1. #1
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    Question If the radius of the small circle is 8 cm, what is the circumference of the...

    I don't know how to do this question, could someone please explain?
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  2. #2
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    Hello, iamanoobatmath!

    I hope you can follow my labelling.
    (I couldn't make a diagram of this one!)

    The vertex of the angle is $\displaystyle A.$
    Draw a line from $\displaystyle A$ through the centers of the circles.

    It first intersects the small circle at $\displaystyle B$,
    the center of the smaller circle is $\displaystyle C$,
    $\displaystyle D$ is the point of tangency of the two circles,
    the center of the larger circle is $\displaystyle E$.

    From C, draw a radius down to the point of tangency, $\displaystyle F.$
    From E, draw a radius down to the point of tangency, $\displaystyle G.$

    We have: .$\displaystyle BC = CD = CF = 8$
    Let $\displaystyle DE = EG = r$

    In right triangle $\displaystyle CFA,\;\angle CAF = 30^o\,\text{ and }\,CF = 8.$
    . . Hence, the hypotenuse $\displaystyle AC = 16.$

    Since $\displaystyle \Delta EGA \sim \Delta CFA$, we have: .$\displaystyle \frac{EG}{EA} \:=\:\frac{CF}{CA}$

    . . That is: .$\displaystyle \frac{r}{r+24} \:=\:\frac{8}{16} \quad\Rightarrow\quad r \:=\:24$


    Therefore, the circumference of the larger circle is: .$\displaystyle 48\pi$ cm.

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