Results 1 to 5 of 5

Math Help - Interior Angles

  1. #1
    MHF Contributor Quick's Avatar
    Joined
    May 2006
    From
    New England
    Posts
    1,024

    Lightbulb Interior Angles

    When I was in 6th grade I came up with an equation for finding the measure of the interior angle of any vertex to a regular polygon.

     180-\frac{360}{n} where  n is the number of sides to the polygon. My equation comes out in degrees.

    But you guys are saying \pi(1-2/n) which comes out in radians.

    Wouldn't it just be easier to say  180-\frac{360}{n} ?

    P.S. I know the equations are the same, one just comes out in radians and the other in degrees; My main question is what is the point of radians?
    Last edited by Quick; June 19th 2006 at 06:49 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by Quick
    P.S. I know the equations are the same, one just comes out in radians and the other in degrees; My main question is what is the point of radians?
    A scientist would say that it is because it reduces the size of numbers or something like that, (stupid reason).

    The mathematical reason is more elegant. In calculus there is relationship because sine and cosine. We say that the derivative of sine is cosine. Now this is ONLY true when the angle is measured in degrees.
    Thus, in radians
    \mbox{derivative of }\sin x=\cos x
    And, in degrees,
    \mbox{derivative of }\sin x^o=\frac{\pi}{180}\cos x^o
    Which one is simpler? The second one of course.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Quick
    When I was in 6th grade I came up with an equation for finding the measure of the interior angle of any vertex to a regular polygon.

     180-\frac{360}{n} where  n is the number of sides to the polygon. My equation comes out in degrees.

    But you guys are saying \pi(1-2/n) which comes out in radians.

    Wouldn't it just be easier to say  180-\frac{360}{n} ?

    P.S. I know the equations are the same, one just comes out in radians and the other in degrees; My main question is what is the point of radians?
    The reasons I would recommend using radians to express angles are:

    1. It is a more natural measure of angle in Mathematics, some of the
    reasons for this have been given by PH, but there are others as well.

    2. One of the common practical uses for trig functions is in scientific
    and engineering computing, where the trig functions by default use
    radian measure. If you habitually use degrees you will end up making
    mistakes if you ever get to write scientific code. One common
    consequence of such errors are that people die (when you bridges
    fall down) or your Mars probe gets lost.

    (same argument applies to using SI units rather than customary)

    RonL
    Last edited by CaptainBlack; June 20th 2006 at 08:13 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,803
    Thanks
    692
    Hello, Quick!

    Here's how I've tried to explain radians and natural logs to my students.


    Someone (a Babylonian?) divided a circle into 360 equal parts and named them "degrees".
    . . Why three-hundred-and-sixty? . . . Who cares? . Does it matter?

    On another planet, say, Gamma Hydra IV in the Delta Quadrant,
    . . the inhabitants might divide a circle into 500 parts and call them " \blacktriangledown\rightsquigarrow\ddots"
    . . (nearest translation).

    These angle measures are totally arbitrary.
    We can create our own system: 1\text{ circle} = 137\text{ gzorns}

    In every case, when we study Calculus, we must insert an "adjustment factor" into the problem
    . . to specify which angle measure we are using.

    Well, almost every case . . . If we use radians, no adjustment is required.
    Evidently, radians are already built into Calculus; they arise naturally.


    What about natural logarithms?
    What's so natural about using such an ugly base: 2.718281828459045 . . . ?

    The number is transcendental (like \pi) and goes on forever, never repeating.
    Who in their right mind would want to raise it to some power?

    In Calculus, logarithmic problems also require an "adjustment factor"
    . . to specify the base we are using . . . except when using base e.

    With natural logs (base e), no adjustment is required.
    So the ugly number e is also built into Calculus.

    Hence, they are called natural logarithms. .(Now you know why!)


    It appears that \pi and e are woven into the very fabric of Mathematics.
    These two numbers turn up in the most unexpected places.
    . . For example: . e^{i\pi} + 1\:=\:0
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,054
    Thanks
    368
    Awards
    1
    One or two Physicists might even say that the reason that radians are preferred is because they are essentially unitless: The measure of an angle in radians is distance/distance (from the circle equation s = r \theta where s is the arc length described by a central angle theta in a circle of radius r). Effectively then radians are unitless, so we can drop the unit.

    (I have investigated what happens when you DON'T drop the radian unit in Physics...the results are fascinating! To me anyway. )

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: June 20th 2011, 01:37 PM
  2. sum of interior angles of a 7-point star.
    Posted in the Geometry Forum
    Replies: 7
    Last Post: April 4th 2011, 09:41 PM
  3. Interior and Exterior Angles of Polygons
    Posted in the Geometry Forum
    Replies: 5
    Last Post: March 1st 2010, 04:58 PM
  4. Interior angles of a Parallelogram
    Posted in the Geometry Forum
    Replies: 2
    Last Post: April 16th 2009, 10:17 PM
  5. Exterior and Interior Angles
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 20th 2007, 10:00 PM

Search Tags


/mathhelpforum @mathhelpforum