Bear in mind that the length of an arc is a part of the circumference of a circle. The formula for the circumference of a circle is: C = 2 × π × r. Remember also that the angle at the centre of every circle is 360 degrees.
Those are two important facts you need to keep in mind when finding arc length.
Now the solution becomes easy.
Find the circumference of the circle (with radius = 20, the length of the needle)
C = 2 × 3.142 × 20 (Use whatever value of pi you were given)
C = 125.68
Find the angle the needle rotates: 82 degrees (This angle is normally called theta (θ) just to confuse you!)
Express this angle as a fraction of 360: 82/360 (There is no harm in reducing it to a decimal if you want: 023)
AND TO FIND THE LENGTH OF THE ARC: simply multiply the fraction (or the decimal 0.23) by the circumference (125.68)
You will get 0.23 × 125.68 = 28.9 cm
θ/360 × 2 × π × r