A circle's center is at (9 ,2 ) and it passes through (5 ,2 ). What is the length of an arc covering (pi ) /3 radians on the circle?

1 Answer
May 24, 2018

(4pi)/3

Explanation:

The distance between the center and any point on the circle is the radius. From the 2 points given in the description, we can determine that the radius r=9-5=4
More formally, the distance between any two points is:
d=sqrt((y_2-y_1)^2+(x_2-x_1)^2)
In this case: r = sqrt((2-2)^2 + (9-5)^2) = sqrt(4^2) = 4

The circumference of this circle is C = 2pir = 8pi

The length of the arc is a portion of the total circumference. Since the total arc around the whole circle is 2pi, the proportion of the arc is (pi/3) / (2pi) = 1/6 of the total circumference.

So, the length of the arc covering pi/3 radians is (1/6)C,
that is:
(1/6) * 8pi = (4pi)/3