### Video Transcript

Determine to the nearest hundredth the length of the arc ππ.

We recall first that this notation means the minor arc connecting points π and π. Thatβs this arc here. Weβve been asked to calculate the length of this arc. We can recall the formula for doing so. If the central angle of an arc in a circle of radius π is π degrees, then the arc length π is given by π equals π over 360 multiplied by two ππ. What this formula is doing is calculating the circumference of the full circle, thatβs two ππ, and then multiplying this by the portion of the circumference we have. If the central angle is π, then the fraction of the circumference that the arc represents is π over 360, as there are 360 degrees in a full turn.

In this circle, the central angle for the arc ππ, thatβs the angle between the two radii connecting the endpoints of this arc to the center of the circle, is 60 degrees. The line segment ππ, which is the diameter of the circle, is of length four centimeters. If the diameter is four centimeters, then the radius is half of this. So it is two centimeters. So, substituting 60 for the central angle π and two for the radius π, we have that the length of the arc ππ is 60 over 360 multiplied by two multiplied by π multiplied by two. 60 over 360 can be simplified to one-sixth by dividing both the numerator and denominator by 60, and two times π times two is four π. So we have one-sixth multiplied by four π. Thatβs four π over six, which simplifies to two π over three.

If we wanted an exact answer or if we didnβt have a calculator, we could leave our answer in this form. But weβve been asked to determine the length to the nearest hundredth. Evaluating this on a calculator gives 2.0943 continuing. And then rounding to the nearest hundredth or two decimal places gives 2.09. The units are centimeters because weβre finding a length and those were the units given for the diameter in the question. So we found that the length of the arc ππ to the nearest hundredth is 2.09 centimeters.