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Math Help - Finding external Circumference of 2 overlapping circles

  1. #1
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    Finding external Circumference of 2 overlapping circles

    I marked the top of the intersection (where the 2 circles meet) "a" and the bottom "b". Then I made a triangle with the midpoint of AB and the centre of the larger circle. I used Sine Law to find the angles and then found the sector length. My final answer came to be 98.89. However my textbook has a different answer, am I doing something wrong here.
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  2. #2
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    Re: Finding external Circumference of 2 overlapping circles

    If you make isosceles triangles using the radii and the length AB you should have a 12, 12, 15 triangle, and a 9, 9. 15 triangle.

    Since in each case you have three sides and an unknown angle, I would use the cosine rule to find the unknown angle.

    Triangle 1:

    \displaystyle \begin{align*} \cos{\theta} &= \frac{12^2 + 12^2 - 15^2}{2\cdot 12 \cdot 12} \\ \cos{\theta} &= \frac{144 + 144 - 225}{288} \\ \cos{\theta} &= \frac{63}{288} \\ \cos{\theta} &= \frac{7}{32} \\ \theta &= \arccos{\frac{7}{32}} \\ \theta &\approx 77.36^{\circ}\end{align*}

    So the length of the major segment for Triangle 1 is \displaystyle \frac{360 - \arccos{\frac{7}{32}}}{360} \cdot 2\pi \cdot 12 \approx 59.12\,\textrm{cm}.


    Now do the same for the second circle and triangle.
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  3. #3
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    Re: Finding external Circumference of 2 overlapping circles

    Hello, castle!

    I marked the top of the intersection (where the 2 circles meet) "A" and the bottom "B".
    Then I made a triangle with the midpoint of AB and the centre of the larger circle.
    I used Sine Law to find the angles and then found the sector length.
    My final answer came to be 98.89.
    However my textbook has a different answer. . What is it?

    Let P be the center of the 12-radius circle,
    . . and Q be the center of the 9-radius circle.

    Now look at \Delta APQ.

    Code:
                      A
                      *
                    *  *
             12   *     *  9
                *        *
              *           *
            * α          β *
        P *  *  *  *  *  *  * Q
                    15

    Note that we have a 3-4-5 right triangle.

    Hence: . \cos\alpha \:=\:\frac{12}{15} \quad\Rightarrow\quad \alpha \:=\:\cos^{-1}(0.8) \:=\:0.643501109\text{ (radians)}

    Then: . \angle APB \:=\:2\alpha \:=\:1.287002218

    And: \text{major }\angle APB \:=\:2\pi - 2\alpha \:=\:4.996183089

    Hence: \text{major arc }\overline{APB} \:=\:12(2\alpha) \:=\:59.95419707


    We have: . \beta \:=\:\tfrac{\pi}{2} - \alpha \:=\:0.927295216

    Then: . \angle AQB \:=\:2\beta \:=\:1.854590432

    And: \text{major }\angle AQB \:=\:2\pi - 2\beta \:=\:4.428594875

    Hence: \text{major arc }\overline{AQB} \:=\:9(2\beta) \:=\:39.85735388


    Therefore, the external circumference is:
    . . 59.95419707 + 39.85735388 \;=\;99.81155095

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  4. #4
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    Re: Finding external Circumference of 2 overlapping circles

    The textbook's answer is 102 cm, but I believe it may have just been a matter of rounding as my answer and Soroban's are very close to the textbook's.
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