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Why we measure angles with Radians

  1. Oct 23, 2011 #1
    Hi guys,

    I had been wondering for a while why it is that we use the radian as the unit of angular measurement in higher sciences and mathematics (calculus, physics, engineering) as opposed to the degree.

    In reviewing the relationship between the degree and radian, I believe that I have developed a decent understanding of them both and why radians are preferred. Perhaps I can receive some confirmation.

    When we measure angles with degrees and radians, we are actually measuring two different quantities.

    The degree is a measure of how wide two rays are opened; the turn or rotation of a complete circle.

    The radian is a ratio of a portion of a circle's arc length to its radius. If the radius of said circle is 1, then the radian is simply just the arc length. So in essence, when we perform angular measurement with the radian, we are not measuring the rotation around the circle, rather we are measuring the distance around the circle. If the radius of the circle is not 1, we can simply multiply radian by the radius to acquire the correct measurement.

    To these ends, it would be more convenient to use radians as opposed to degrees.

    Is this correct?
  2. jcsd
  3. Oct 23, 2011 #2


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    In a nutshell yes.
    A degree has a unit - degrees.
    A radian is dimesionless since it is a ratio - arc length/radius
  4. Oct 23, 2011 #3


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    Radians are also the natural choice for expressing angles in calculus. It is the only choice for which the derivative of sin(θ) is cos(θ). Any other unit of measurement would require an extra "fudge factor". It is similar to the reason that e is the "natural" base for logarithms and exponentials.
  5. Oct 24, 2011 #4


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    If we have a small angle x we have
    sin(x)~a x
    where ~ means approximately equal.
    In fact a=sin'(0) in calculus.

    This a is used all over the place in many formula so it is convenient to use radians so that a=1 to simplify such formula.
  6. Oct 25, 2011 #5
    Because they are easily identified with real number , so we can identify sine and cosine as real valued functions , and apply Calculus to them easily .
  7. Oct 25, 2011 #6
    that is the natural way, a radian measures the lenght of the arc of the circle. 2pi radians is the lenght of the circumference of a circle of radius 1 and so on.
  8. Oct 25, 2011 #7
    Because we have this equation

    It follows that

    But if

    Does it make any sense?
    Last edited by a moderator: Apr 26, 2017
  9. Oct 25, 2011 #8


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    I've always thought that it is because radians tie naturally to the circle and don't require any further definitions (that is, C= pi * D or C = 2 * pi * R, and the circle is 2 * pi radians) whereas degrees are a purely arbitrary man-made set of units, so some math flows MUCH more naturally with radians than degrees.

    That said, I sure do like degrees better as a measure of an angle. I mean, you tell me 20 degrees and I have a good idea what you're talking about but you tell me .35 radians and I'll likely look at you like your a Martian. :smile:
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