Are Angles Dimensionless?

Date: 08/31/2003 at 12:30:32
From: Christopher
Subject: Dimensional inconsistencies

In the equation 

  arc length = r*theta

the numbers seem to work out fine, but the dimensions seem to be
completely incorrect. 

In any circle there is a specific angle where the radius is equal 
to the arc length. This angle is defined as one radian and is
approximately equal to 57.29 degrees. 

From this it can then be said that for every length of arc that is
equal to the radius there is 1 radian of angle subtended. For any
sector the angle will be so many lots of 1 radian, and thus the arc
length will be the same number of times of the radius. From this you
can then write:

 arc length / angle (in radians) = radius / 1 radian

It can now be said that:

  arc length = angle (in radians) * radius / 1 radian

This is not the same as the equation 

  arc length = r * angle (in radians) 

Furthermore the r*theta equation seems to imply that the angle is 
dimensionless (has no units!). This is obviously incorrect, as it 
seems to imply that an angle can be measured with numbers. 

You have to have some way of measuring an angle just like you do with
mass, or with length. Units are simply a standard measure of some
parameter, which tell you what you are dealing with, whether it is
length, mass, angle or whatever. The numbers just tell you how many of
them you have. How you can say that the angle measured has no units
goes against everything I know.  But I am not arrogant enough to say
the whole world is doing it wrong and I am doing it right.  Can you
explain to me what is going on here?

Date: 08/31/2003 at 23:16:14
From: Doctor Peterson
Subject: Re: Dimensional inconsistancies.

Hi, Christopher.

Angles are indeed dimensionless; the reason is that an angle is
measured as the ratio of arc length to radius (meters per meter, say),
which cancels out any units. Other angle measurements, such as
degrees, are also dimensionless, though they are defined by different
ratios, such as the ratio of the arc length to 1/360 of a circle,
which results in the need for a multiplier. It is for this reason that
we still need to name the unit, radians, rather than leaving it out
entirely. Saying "radians" specifies the way in which the angle is
measured; but it is still dimensionless and can be ignored as long as
you know that radians are the correct method for the calculation being
done.

Since this is a tricky issue, I did a web search for references to 
support my opinion. Here are some:

  Why Use Radians instead of Degrees?
  http://mathforum.org/library/drmath/view/54181.html 

  All angle measures can be said to be dimensionless. A radian is
  the ratio of an arc length to a radius, and the ratio of two
  lengths is dimensionless. A degree is 180/pi radians, and the
  constant 180/pi is dimensionless, so a degree is also
  dimensionless.

  Trig functions are also by nature dimensionless. They can be
  defined as ratios of sides of a right triangle. Again, the ratio
  of two lengths is a dimensionless quantity (a "pure number").

  Glossary of Mathematics
  http://westview.tdsb.on.ca/Mathematics/R.html 

  Radian Measure: A dimensionless unit of angle measure (as opposed
  to degree which has a dimension). An angle of one radian (1 rad)
  causes the ratio of the length of the arc of the circle's
  circumference to the radius of the circle to be one. Since this
  is a ratio the radian is not a unit and it is dimensionless. We
  have two different angle measures (pi radians = 180 degrees)
  since it is very troublesome to integrate or differentiate trig
  functions whose arguments are in degrees. One radian
  approximately equals 57.3 degrees. 

  One of the real values of radians is that it is a non-unit "unit"
  (since it is the ratio of an arc length to the radius of a circle).
  That is, in any calculation using an angle measure, using radians
  does not introduce a new unit into the process to muddy the
  situation. If you are putting an angle into a trig function, it
  doesn't matter whether you use radians or degrees, since, for
  example, sin 30 deg. = sin pi/6 rad. = sin pi/6 (no unit). IF you
  are using an angle value in any other calculation, however, using
  degrees often produces non-sense. For example, evaluate the
  expression x^2-2x-4 for x being 30 deg. or pi/6 radians (or,
  therefore, just plain pi/6 with no unit). With degrees, the result
  is non-sense -- you get 900 square degrees - 60 degrees - 4. With
  radians, the non-unit "unit", the result is a pure, unitless
  number. Evaluating Taylor polynomials in order to get approx.
  values of the trig functions for any angle, then, requires
  radians to make any sense whatsoever. Calculators use similar
  methods to produce trig values, so if you ask your calculator for
  sin 30 degrees, it internally converts the angle to radians, uses
  that value in an algebraic expression that it has stored in a
  memory location that your "sin" key accesses, then produces the
  appropriate UNITLESS value for sin 30 degrees. Likewise in any
  other formula - in math, science, engineering, architecture, or
  whatever - if you have to use an angle in most any way other than
  in a trig function, radians are required and degees produce non-
  sense.

I disagree with this as to degrees; I would say the degree is still 
dimensionless, but is a different unit (scaling). But the radian is 
generally assumed, so that the degree requires conversion. For 
example, the Taylor series for the sine, as usually given, assumes 
that the sine function takes an angle in radians; but one could write 
a Taylor series for the sine of an angle in degrees, which would 
simply have different coefficients. There is no inherent difference, 
contrary to what this author says; it is just a rescaling.

This is why we do include the radian as a unit in calculations, as a 
reminder of the correct scaling; but it can be ignored where 
necessary.

  Introduction to the Aerodynamics of Flight
  http://history.nasa.gov/SP-367/appendb.htm 

  The measure of the central angle of a circle is defined as the
  ratio of the subtended arc of the circle divided by the radius,
  that is, a ratio of two lengths. Thus, this measure is
  dimensionless but is assigned a special name of radians.
  Additionally one may express the angle in degrees by noting that
  an angle of 1 radian equals about 57.3°. The fact that both
  radian measure and degree measure are dimensionless means that
  the numerical value of an angle does not change from one system
  of units to another.

I'm not sure what that last sentence means, but otherwise I agree.

  The Three Classes of SI Units and the SI Prefixes
  http://physics.nist.gov/Pubs/SP811/sec04.html 

  The SI supplementary units are now interpreted as so-called
  dimensionless derived units (see Sec. 7.14) for which the CGPM
  allows the freedom of using or not using them in expressions for
  SI derived units. (Footnote 3) Thus the radian and steradian are
  not given in a separate table but have been included in Table 3a
  together with other derived units with special names and symbols
  (see Sec. 4.2.1). This interpretation of the supplementary units
  implies that plane angle and solid angle are considered derived
  quantities of dimension one (so-called dimensionless quantities
  - see Sec. 7.14), each of which has the unit one, symbol 1, as
  its coherent SI unit. However, in practice, when one expresses
  the values of derived quantities involving plane angle or solid
  angle, it often aids understanding if the special names (or
  symbols) "radian" (rad) or "steradian" (sr) are used in place of
  the number 1. For example, although values of the derived
  quantity angular velocity (plane angle divided by time) may be
  expressed in the unit s^-1, such values are usually expressed in
  the unit rad/s.

This is "official"! The radian is optional, and can be replaced by 1.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/