# A circle's center is at (4 ,6 ) and it passes through (3 ,1 ). What is the length of an arc covering (pi ) /3  radians on the circle?

Feb 8, 2016

The arc length is $\frac{\sqrt{26}}{3} \pi$.

#### Explanation:

First of all, you need to compute the radius.

If you center is at $\left(4 , 6\right)$ and an arbitrary point on a circle is $\left(3 , 1\right)$, we can compute the radius as follows:

$r = \sqrt{{\left(4 - 3\right)}^{2} + {\left(6 - 1\right)}^{2}} = \sqrt{1 + 25} = \sqrt{26}$

Now, the the length of an arc covering the whole circle would be equivalent to the perimeter of a circle, $2 \pi r$.

In your case, you would like to compute an arc covering $\frac{\pi}{3}$ radians instead of the whole $2 \pi$ (equivalent to ${60}^{\circ}$ which is $\frac{1}{6}$ of the whole circle).

Thus, your arc length is $\frac{\pi}{3} r = \frac{\sqrt{26}}{3} \pi$.