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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.

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:

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

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

Now do the same for the second circle and triangle.

Re: Finding external Circumference of 2 overlapping circles

Hello, castle!

Quote:

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 $\displaystyle P$ be the center of the 12-radius circle,

. . and $\displaystyle Q$ be the center of the 9-radius circle.

Now look at $\displaystyle \Delta APQ.$

Code:

` A`

*

* *

12 * * 9

* *

* *

* α β *

P * * * * * * * Q

15

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

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

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

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

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

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

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

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

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

Therefore, the external circumference is:

. . $\displaystyle 59.95419707 + 39.85735388 \;=\;99.81155095$

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.