Why radian has no dimension?

Answer 1

The simplest explanation is that a radian is defined as the distance an angle would subtend along the circumference of a circle of radius "r" divided by the radius "r." If we call the subtended arc "a," then both "r" and "a" are measured in length units, which of course cancel out when the ratio "a" over "r" is taken.

The easiest way to understand this is probably through an example. Take an angle of 30 degrees. If we construct a unit circle (radius = 1 unit) centered at the vertex of the angle, then we know the circumference of the circle is 2(pi)r = 6.283 units. We also know that the circumference of the circle is 360 degrees by definition, so a 30 degree angle subtends 30/360, or 8.3 percent of the circumference of 6.283 units, or 0.524 units. Thus, the angle in radians is the length of the subtended arc, 0.524 units, over the length of the radius, 1 unit, or 0.524 units/1 unit = 0.524 (with the units dimension canceling out).

Since by definition, circumference = 2(pi)(r) = 360 degrees, it is easy to see that an arc of "r" units (that is one radian) = 360/(2(pi)) = 57.3 degrees and that there are 2(pi), or 6.28 radians in 360 degrees. If one radian = 57.3 degrees, then the 30 degree example above should equate to 30/57.3 = 0.524 radians, just as was shown.

Bottom line is that an angle in radians defines a given subtended arc distance for a circle of any radius measured in any units, as long as it is centered at the vertex of the angle. If a circle of radius 57 hoozits is centered at the vertex, then an arc of length "57 times the angle in radians" hoozits is subtended. Similarly, if a circle of radius 23 cm is centered at the vertex, then an arc of length "23 times the angle in radians" cm subtended. In either case, the subtended angle in degrees is the angle in radians times 360/((2)(pi)).