<P> Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc . One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle . More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s / r, where θ is the subtended angle in radians, s is arc length, and r is radius . Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ . </P> <P> As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted . When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol ° is used . </P> <P> It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr / r, or 2π . Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180 / π degrees . </P> <P> The relation 2 π rad = 360 ∘ (\ displaystyle 2 \ pi (\ text (rad)) = 360 ^ (\ circ)) can be derived using the formula for arc length . Taking the formula for arc length, or l a r c = 2 π r (θ 360 ∘) (\ displaystyle \ ell _ (arc) = 2 \ pi r \ left ((\ frac (\ theta) (360 ^ (\ circ))) \ right)). Assuming a unit circle; the radius is therefore one . Knowing that the definition of radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, we know that 1 = 2 π (1 rad 360 ∘) (\ displaystyle 1 = 2 \ pi \ left ((\ frac (1 (\ text (rad))) (360 ^ (\ circ))) \ right)). This can be further simplified to 1 = 2 π rad 360 ∘ (\ displaystyle 1 = (\ frac (2 \ pi (\ text (rad))) (360 ^ (\ circ)))). Multiplying both sides by 360 ∘ (\ displaystyle 360 ^ (\ circ)) gives 360 ∘ = 2 π rad (\ displaystyle 360 ^ (\ circ) = 2 \ pi (\ text (rad))). </P>

In the radian system of angular measurement what is the measure of one revolution