Question

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?

Answer 1

The arc length is `sqrt(26)/3 pi`.

Explanation:

First of all, you need to compute the radius.

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

`r = sqrt((4-3)^2 + (6-1)^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, `2pi r`.

In your case, you would like to compute an arc covering `pi/3` radians instead of the whole `2pi` (equivalent to `60^@` which is `1/6` of the whole circle).

Thus, your arc length is `pi/3 r = sqrt(26)/3 pi`.