A circle's center is at (2 ,4 ) and it passes through (3 ,1 ). What is the length of an arc covering (13pi ) /12 radians on the circle?

1 Answer
Feb 18, 2016

arc length = radius x arc angle
radius = sqrt(10)
arc angle = (13pi)/12

arc length = sqrt(10)((13pi)/12) ~= 10.76

Explanation:

First find the radius, r, using the equation of a circle:
(x-x_0)^2+(y-y_0)^2=r^2

The circle's centre is (2,4), so x_0=2 and y_0=4
The circle passes through (3,1), so at this point x=3 and y=1.

Inputting all this into the circle equation:
(3-2)^2+(1-4)^2=r^2
(1)^2+(-3)^2=r^2
1+9=r^2=10
Therefore: r = sqrt(10)

Now we can find the arc length using:
arc length = radius x arc angle

arc angle = (13pi)/12

arc length = sqrt(10)((13pi)/12) ~= 10.76