Degrees Suck!

It seems to be common, at least in my country, to measure angles in a unit known as "the degree". This is unfortunate as the degree in truly a primitive and barbaric unit of angular measure. Supposedly, its origins lie in the fact that ancient cultures divided the year into 360 days so the degree divides the circle into 360 segments. This is quite a poor way to do mathematics.

There is a solution! The Radian is a natural unit of angular measure. However, it is not often introduced until the student's mind has already been corrupted by exposure to degrees. This is highly unfortunate in that it encourages the student to think in terms of degrees when trying to understand radians.

The fundamental problem that both units are trying to address is providing a way of measuring how "large" angles are. Suppose we have an angle:

http://i817.photobucket.com/albums/z...oned/angle.jpg

How are we to determine how large this angle is? Both methods begin with superimposing a circle over the angle:

http://i817.photobucket.com/albums/z...ned/circle.jpg

Doing so provides a beautiful way to answer our question which the method of degrees completely overlooks. We have the equation:

c = 2πr

This says that the circumference of a circle is equal to two times pi times the radius of the circle. Let us normalize the circle so that it has radius 1. We can do this by dividing both sides by the radius.

c/r = 2π

We see that a circle of radius 1 has a circumference of 2π. The idea of radians is to declare that we will measure angles by the length of the portion of the circle which is contained within the angle after the circle is normalized to radius 1. So a full circular angle (that is turning around in a full circle) is 2π radians. A semi-circular angle is π radians and a quarter-circular angle is π/2 radians. In general, if an angle goes around a circle x times, then it is 2πx radians. So an angle that goes around a circle 1/8 times is π/4 radians.

Another indication of the elegance of radians is in the simplicity of measuring them. Whereas one needs to carry around a circle with 360 marks around it to measure degrees, one needs only a length of string to measure a radian. Simply lay the string along the portion of the edge of the circle that is contained within the angle. Then see how many times you can mark off the radius of the circle along the same length of string. That is the "size" of the angle in radians. Granted, one could measure the degrees in this manner too but only by measuring it in radians and then converting to degrees.

Finally, suppose you want to know how long a circular arc is for some angle. If the angle is specified in radians, you may simply multiply the angle times the length of the radius.

arclength = αr

We see that we recover the formula for the circumference of the circle when α = 2π.

http://i817.photobucket.com/albums/z..._you_know2.jpg

"Turn and fly heading 3/4π rad"

...no, that doesn't flow well at all...

0.750 pirad; sounds cool to me.

Well from a practical, non-mathematical perspective, you would probably want to do something like change units to degrees because factors of 360 are easy to work with but the introduction to angular measure should be radians. You should just think of 1 degree as π/180 radians. All I'm saying is that the radian is the natural unit of circular measure and I don't think it's pedagogically right to introduce a contrived unit before a natural one.

Actually, even this is a little contrived. It shouldn't be written in terms of π but in terms of τ = 2π. Then we have c = τr. We just normalize on the radius again and get c/r = τ. Then we just have that the measure of an angle could be the number α such that τα is equal to the length of segment of the unit circle traced by the angle. From the formula c = τr we see that we can see that we must have 0 < α < 1 as long as the angle only goes around the circle once. The number α is just how many times the angle goes around the circle. 0 is none and 1 is once in the positive direction. So then one quarter or 1/4 or .25 or even 25% would then refer to one quarter turn. I wouldn't want to write a proof with it but that kicks ass over the degrees for being easy to use.

I didn't exactly understand the part about string. Is that supposed to be some elementary school trick? I was never taught about radians, although I do they are the "real" measuring unit when it comes to angles. It can be quite frustrating when working with MathCad, because it outputs a lot of things in radians.

tl;dr more people write clever explanations.

Imagine a string going from the center of a circle to the edge. So the string denotes the radius. Now imagine wrapping that same radius-length piece of string around the circumference of the circle. That's a radian.

How many of the strings (radians) do you need to reach the other side of the circle? pi radians. To get back to where you started? 2pi radians. That's why the circumference of a circle is 2 * pi * radius. Radians are based on the radius.

Basically think of any angle as describing a segment of a circle, and then radians measure how many radii fit into along the edge of the circle.

It's very natural and intuitive.

The only problem is that there isn't an even (or even rational) amount of radians in a circle, which is probably why they don't teach them to 4th graders.

Uhm, where the hell did you get the info that the 360 has anything to do with the number of days in a year? They chose 360 because it's divisible by a lot of things evenly: 2, 3, 4, 5 ,6 ,8, 9, 10, 12, 15, 18, 20, 30, 45, 60, 90, and 180.

I don't see why it's a problem for students that the number of radians in a circle isn't nice and simple. You don't usually need to measure the angle of a circle...

Quote:

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Originally Posted by Spartiate

The only problem is that there isn't an even (or even rational) amount of radians in a circle, which is probably why they don't teach them to 4th graders.

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I don't think that's a valid complaint. If I'm not wrong, they teach the formula c = 2πr to fourth graders. Then it's just division to get that half of a circles angle is 2πr/2 = πr. But we can then divide both sides by r.

Honestly, if they're not old enough to learn that if two numbers are equal before division, then they're equal after, then they're too young for math. If there is some reason that they are biologically unable to grasp this then they're too young to learn division to begin with. They're just learning an algorithm but not getting a feel for what that algorithm accomplishes. This is why you end up with a lot of people that can't do basic arithmetic to count change and shit. They may have once known how to divide but they don't know what division is.

So now we have that the radius doesn't matter in this equation (as long as not zero) because it occurs on both sides of an equation and the angle that goes halfway around the circle is π. You can do a similar exercise with any other angle. Just multiply any fraction, α, by 2π and that's the angle that goes α of the way around the circle. That's super easy.

The thing is degrees are generally taught in basic geometry to measure angles within different polygons and shapes. Using rads for that would be incredibly tedious.

I don't think I was taught about pi and the circumference equation until high school (which would be the equivalent of 7th grade in the US).

Ah, I see, it all makes a lot of sense then.

Quote:

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Originally Posted by Spartiate

The thing is degrees are generally taught in basic geometry to measure angles within different polygons and shapes. Using rads for that would be incredibly tedious.

I don't think I was taught about pi and the circumference equation until high school (which would be the equivalent of 7th grade in the US).

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Straight line: 1 pirad, 180 degrees.
Right angle: 1/2 pirad, 90 degrees.
Equilateral angle: 1/3 pirad, 60 degrees.
Pentagonal angle: 2/5 pirad, 72 degrees.
Septagonal angle: 2/7 pirad, 51.4285714 degrees.

Degrees are less tedious for geometry? Really..?

I think that's just familiarity speaking there; looks to me like it's a whole lot more stuff to learn.