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If $\angle ACB = 40^{\circ}$(see figure), and the area of the circle is $81\pi$, how long is the arc $ABC$?

enter image description here

This is how i have approached this problem.

First of all we know the area of cricle is $81\pi$ therefore the radius is $9$.

We can use this to find the circumference which would equal to $18\pi$.

In order to find the arc length we can use arc length formula which is as follows

$$\text{arc length} = \frac {x}{360^{\circ}} \cdot \text{circumference}$$ $$= \frac{40}{360} \cdot 18\pi$$ $$= \frac 19 \cdot 18\pi$$ $$= 2 \pi$$

So i end up getting $2\pi$ but that is not the correct answer because the book is saying that $\angel AXB = 80^{\circ}$. This can only be true if it were a central angle. But this figure clearly does not show that AXB is a central angle or am i wrong? Because my understanding of a central angle is that it would look something like this

enter image description here

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    $\begingroup$ I must apologize for this, its my mistake. The reason my answer is wrong is because in order to find an ARC LENGTH you require the central ANGLE. In this question we are given the angle of the inscribed circle which is 40 degrees and its central angle would be 2 times that which would be 80 degrees hence the final answer would be 80/360 * 18 pi and the correct answer would be 4pi. I am extremely sorry for this! How do i delete this post? $\endgroup$ Commented Sep 22, 2013 at 5:03
  • $\begingroup$ write it as an answer and accept your own answer please so e do not have lot of unanswered questions and well done for figuring out $\endgroup$
    – Mr. Math
    Commented Sep 22, 2013 at 5:31

1 Answer 1

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The reason my answer is wrong is because in order to find an ARC LENGTH you require the central ANGLE. In this question we are given the angle of the inscribed circle which is $40 ^{\circ}$ and its central angle would be 2 times that which would be $80^{\circ}$ hence the final answer would be $\frac{80}{360} \cdot 18\pi$ and the correct answer would be $4\pi$.

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