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Angle Measures

October 2, 2011

Suppose nobody had ever thought of measuring the size of an angle, and someone asked you “How can I describe the size of an angle?” What approach might you take in answering this question?

You might start by arbitrarily picking some angle, any angle, such as angle ABC in the image below, and call its measure “1″. All other angles could be

referred to as multiples or fractions of this angle. Therefore, angle ABD would have a measure of 2, and angle ABE would have a measure of 3:

As you consider larger and larger angles, each a multiple of the original angle ABC above, and thinking about the measure of each, you will eventually get to an angle which is so large that it has created a full circle. However, did that last angle end exactly along AB, or did it actually produce an angle greater than a full circle?

In the image above, clearly the next multiple of the original angle will not land on point A, but instead will fall somewhere between points A and C. Wouldn’t it be better to have the angle measure of a full circle be a nice round number? This would make it easier to identify when an angle has grown to a measure greater than a full circle, and might also make it easier to identify common fractions of a circle, such as half and quarter circles. Unfortunately, the chances of this happening are poor when an arbitrary starting angle, such as angle ABC above, is used.

“Anglons”

With this idea in mind, lets figure out what a good “base” angle measure might be. If we create an angle measure for a full circle that is evenly divisible by 2, 4, 5, and 8, it would be easy to find the angle measure of many commonly used fractions of a circle.

The least common multiple of these desirable divisors is 40… so we could describe a full circle as having a measure of 40 “Anglons” (a term I just made up to describe this particular approach to measuring angles). Half a circle would therefore have a measure of 20 “Anglons”, a quarter circle would measure 10 “Anglons”, and an eighth of a circle would measure 5 “Anglons”.

Measuring many commonly used angles would be pretty easy, and the standard 1 “Anglon” angle could be obtained by dividing a circle into 40 equal parts.

Degrees

This is probably the sort of thinking that lead to the use of 360 Degrees as the most commonly used angle measure of a full circle today. Note that 360 is exactly nine times the number proposed for “Anglons” above, which makes a one Degree angle equal to exactly one ninth of a one “Anglon” angle, and therefore more practical when describing small differences in angles.

So, Degrees offer 9 times the resolution of “Anglons”, and allow the angle measure of a circle to be easily divisible into 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, or 360 parts. Degrees have proven to be a useful and practical measuring system for angles for much of our recorded history. Wikipedia has an interesting history of Degrees, along with further speculation about how the number 360 came to be used to represent the Degree angle measure of a circle, as well as descriptions of some additional ways of measuring angles that are not described here.

I suspect that the use of 360 Degrees as the angle measure of a circle was probably arrived at by just as arbitrary a process as the one I followed to come up with “Anglons” above. That does not detract from their usefulness or practicality in any way… but there is another commonly used unit of angle measure, one that has a slightly less arbitrary basis.

Radians

It would be nice to have an angle measure be a real “measure” of some physical attribute of an angle… instead describing an angle solely as a fraction of a circle. What measurable property of angle could meet this goal?

The sides of an angle can extend to any arbitrary distance, including infinity, and do not tell us anything about how “wide” the angle itself is. However, if we construct a circle (using any radius we wish) using the vertex of the angle as the circle’s center, we can see that the angle intercepts a section of the circle, an “arc”. The larger an angle is, the longer this intercepted arc will be. The arc length provides us with a good angle measure.

There is a problem though… the length of an arc intercepted by an angle will depend on the radius of the circle used – the greater the radius, the longer the arc will be. In the illustration below, you can see that arc BD is half the length of arc AE, even though they are both intercepted by the same angle:

To resolve this problem, we can decide to measure all arc lengths at the same (arbitrary) distance from the vertex of the angle, say 1 unit. Or, we can also divide the arc length by the circle’s radius to arrive at an angle measure that does not depend on the radius of the circle used:

\dfrac{0.278\pi}{2}=\dfrac{0.139\pi}{1}=0.139\pi

This value is called the Radian measure of an angle, and corresponds to the length of the arc intercepted by the sides of the angle at 1 unit away from the vertex of the angle – as you can see in the image above. The unit descriptor “Radian” could therefore perhaps be more fully described as “Units of Arc length per Unit of Radius”, which is a dimensionless quantity, but the term Radian is a bit less long-winded.

Measuring angles using Radians produces angle measures that correspond to the intercepted circumference of a circle with a radius of 1. Therefore, the measure of the angle of a full circle using Radians will be the same as the circumference of a circle with a radius of 1:

2\pi r\:=\:2\pi\cdot 1\:=\:2\pi

The measures of smaller angles can be determined by using the proportion of a 1 unit radius circle’s circumference that they intercept:

Converting between Degrees and Radians

To convert from one angle measure to the other, recall that both angle measures MUST represent the same proportion of a whole circle:

\dfrac{Part}{Whole}\:=\:\dfrac{Degree\:Measure}{360\:Degrees}\:=\:\dfrac{Radian\:Measure}{2\pi\:Radians}

So, to convert from one measure to the other, divide the measure you have by the measure of a whole circle using the same units, then multiply this proportion by the measure of a whole circle in the desired units.

For example, to convert 30 Degrees into Radians:

(Proportion)\cdot (Whole\:Circle)\:=\: \left(\dfrac{30\:^\circ}{360\:^\circ}\right)\cdot (2\pi\:Radians)=\dfrac{\pi}{6}\:Radians

or to convert \dfrac{\pi}{6}\:Radians into Degrees:

(Proportion)\cdot (Whole\:Circle)\:=\: \left(\dfrac{\dfrac{\pi}{6}\:Radians}{2\pi\:Radians}\right)\cdot (360\:^\circ)\:=\:30\:^\circ

Commonly used angle measures

It is useful to be equally comfortable with both the Degree and Radian measures of common angles. Common angle measures in both systems are:

Degrees Radians
0 0
30 \dfrac{\pi}{6}
45 \dfrac{\pi}{4}
60 \dfrac{\pi}{3}
90 \dfrac{\pi}{2}
120 \dfrac{2\pi}{3}
135 \dfrac{3\pi}{4}
150 \dfrac{5\pi}{6}
180 \pi
270 \dfrac{3\pi}{2}
360 2\pi

Why bother with two angle measurement systems?

Both Degrees and Radians have their advantages and disadvantages as angle measures.

Angle measures expressed in Degrees have the advantage of often being nice round numbers (30 Degrees or 135 Degrees), whereas angle measurements expressed in Radians frequently have a pesky Pi in them \left( \dfrac{\pi}{6}\quad or\quad\dfrac{3\pi}{4}\right) .

Angle measures expressed in Radians are easier to convert into arc lengths. Suppose a 15 inch tire on an automobile has just rotated 45 Degrees (otherwise known as \dfrac{\pi}{4}\:Radians)… how far did the car just travel? If we use the Radian angle measure, we multiply it by the Radius of the tire to get the arc length:

\dfrac{\pi}{4}\:Radians\cdot15\:inches=11.781\:inches

Whereas if we use the Degree angle measure, we calculate its proportion of the circle, then multiply that proportion by the circumference of the circle:

\dfrac{45^\circ}{360^\circ}\cdot 2\pi\cdot 15\:inches=11.781\:inches
which simplifies to
\dfrac{45}{180}\cdot \pi\cdot 15\:inches=11.781\:inches

So, the Radian measure of an angle is more convenient if we need to convert angles to arc lengths frequently (as happens often in Physics), while the Degree measure of an angle is more convenient if we are navigating or building a woodworking project.

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4 Comments leave one →
  1. October 2, 2011 12:49 pm

    I’ve heard of an alternative reason for 360 degrees which I found interesting.

    The Babylonians had 360 days in their official year (and 5 days out of the year set aside for religious holidays) which they presumably chose because it was fairly easy to divide nicely into months, which are themselves approximately the length of one lunar cycle.

    I don’t know if that’s the actual way degree measures of angles was invented, and I certainly like your alternate theory.

  2. October 2, 2011 1:09 pm

    Nice post Whit. Khan Academy watch out!
    Michael ( http://www.mainearchitecture.blogspot.com )

  3. Matt E permalink
    October 4, 2011 10:34 am

    Have you heard of the tau “movement”? I’d be curious as to your thoughts.

    http://tauday.com

    • October 4, 2011 11:13 am

      Up until now, no! After reading through the article, and initially wondering “why bother advocating for yet another constant, when it is an integral multiple of a familiar one”, the idea of using tau is growing more appealing by the minute. I agree that it makes the radian measures of angles much more consistent with one’s intuitive understanding of an angle measure as a proportion of a circle – something that is very relevant to my musings above, and appeals greatly to my memorization-avoidance tendencies. With younger students, I might be inclined to avoid introducing a new greek letter, and settle for keeping 2pi as my constant as a means of achieving a similar level of parallelism. However, with older students (17+ ?) – and particularly ones who are comfortable with math, introducing tau shouldn’t slow them down.

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