Degrees are a convention. Three hundred and sixty of them to a circle, because 360 divides nicely and the Babylonians liked base 60. Radians are not a convention at all; they come out of the geometry.
one radian = the angle where the arc equals the radius
Put a radius of 8 and an angle of one radian into the arc length calculator and the arc is exactly 8.000. Put radius 1 and one radian and the arc is 1.000. The unit is defined so that this works, which makes the arc length formula the simplest possible statement.
arc = r × θ, in radians only
What degrees cost you
In degrees the same formula needs a conversion factor bolted on: arc equals r times the angle times π over 180. At radius 8 and 75 degrees the calculator gives an arc of 10.472, and it reports the angle as 1.309 radians, which is where that figure actually comes from.
Every formula involving angle and length carries the same tax. Sector area, angular velocity, the small angle approximation and every derivative of a trigonometric function are all stated in radians, and using degrees means carrying π over 180 through the whole calculation or converting at the start.
The derivative is the clearest case. The derivative of sine is cosine only when the angle is in radians. In degrees it is cosine multiplied by π over 180, which is the kind of stray constant that makes an entire subject harder than it needs to be. Radians were not invented to be awkward; they were adopted because every alternative is worse once calculus starts.
Why one radian is such an ugly number
One radian is 57.30 degrees, and a full circle is 2π of them, roughly 6.283. Neither is a round number in the decimal sense, and that is the price of the definition: you cannot have both a whole number of parts in a circle and an angle unit that makes arc equal radius times angle.
Degrees chose the round circle. Radians chose the clean formula. Engineering and navigation mostly use degrees because the numbers are human-sized and divide well; mathematics and physics use radians because the formulas do not survive the alternative.
The unit error is a real one
Because the two are interchangeable in the input box and not in meaning, mixing them produces answers that are wrong by a factor of about 57, which is large enough to be caught and quiet enough to slip through a long calculation.
The arc length page guards against the obvious version: enter 75 with the unit set to radians and it refuses, because 75 radians is nearly twelve full turns and almost certainly means 75 degrees. Most software will not warn you, and a trigonometric function in a spreadsheet or a programming language takes radians by default, which is the single most common cause of a trigonometry answer that is confidently and badly wrong.
For the area version of the same geometry, sector area takes an angle in degrees and reports both the area and the arc: at radius 10 and 60 degrees it gives 52.360 square units, an arc of 10.472 and exactly a sixth of the circle, which is the sanity check that a 60 degree sector should pass.
For the other place where a right angle does practical work, see squaring up with 3-4-5.