Find the length of arc s, if r = 10 and 0(with a line through it) = pi/6

Expert Answers
hala718 eNotes educator| Certified Educator

The arc S is drawn between two lines with angle = pi/6

The arc represents a ratio of the circumference. This ratio would be the same ratio between the angles.

The total angle of the circle is 2*pi = 360 degrees

Then the angle pi/6 represent a ratio (R) of (pi/6)/ (2*pi )

==> R= (pi/6 )/ 2*pi= 1/12

Now we will calculate the circumference:

 C = 2Pi*r   (r is the radius)

C = 2(10)*pi= 20*pi

Then the arc = 20*pi * R

                  = 20pi* (1/12)= 5pi/3

OR, simply the formula is:

 S = rc  where (r) is the radius and C is the central angle of the arc in radians.

neela | Student

The circumference of a circle for the angle of 2pi is 2pi*r.

The arc length l is also in the same proportion to the angle it subtends at the centre of the circle. So by ratio, the angle theta and the full round angle 2pi of the circumference at the centre  are in  theta:2pi or theta/(2pi). So the arc length l of the arc theta is, therefore,  given by l = (theta)*r/(2pi) , where r is the radius

In this case, theta = pi/6 and r = 10. Therefore, l = (pi/6)*10/(2pi) =  5/6 is the arc length