That depends upon what type of measurement you are looking for and what kind of an angle you are given. If you are looking for the measure of the arc in degrees AND the angle is a central angle, then the degree measure of the arc is the same as the degree measure of the central angle. If you are looking for the measure of the arc in degrees AND the angle you are given is inscribed, then the arc is double the degrees of the inscribed angle. If you are looking for the measure of the arc in units like inches or feet or meters, there is a formula for that:
where the angle theta is the degree measure of the central angle, and r is the radius of the circle. You have to know the radius to use this formula if what you are looking for is the arc length. If you don't know the radius this formula will be useless to you since you can't solve an equation with more than one unknown. Those are the ones you probably need to be most concerned with.