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## High school geometry

# Radians & degrees

Sal discusses the general approach to converting between radians and degrees and vice versa. Created by Sal Khan.

## Want to join the conversation?

- Can you possibly have negative angles?(38 votes)
- Negative angles are clockwise angles. (Counterclockwise is positive)(72 votes)

- If pi continues forever, how can we use it to define answers? That would mean every answer we get would continue on forever, but we shorten pi and thus makes none of the math we do with pi actually 100% true but rather an estimated amount. I don't even understand the concept of pi honestly. Can someone explain to me?(27 votes)
- Take a measurement of a length of anything, we won't get an exact whole number. A pencil said to be 8 cm long may be 8.000034 cm for example. We are always estimating because the exact amount is almost never needed, and we take as accurate a measurement as required. So, every answer may continue on forever, but what we estimate, is what we need practically.

Theoretically in math, since we always use rational numbers most of the time, an irrational number like pi is often confusing as it does not provide a definite rational answer. Instead we have to estimate to the accuracy required for the situation. If you want to find the circumference of a random cart wheel, you dont need accuracy. When you find the circumference of a rocket, you may need more accuracy.

But of course, theoretically we can still get a definite answer if we just dont expand π and leave it as π. Circumference of a circle of diameter 3 is 3π. This gives you a perfect theortical answer. Otherwise it would be 3*3.14..... and as you said, it is not a perfect defined answer and is not theoretically accurate.

Hope this helps!

- Super7SonicX(54 votes)

- Is there any kind of notation for radians?(21 votes)
- Yes, there is, though it is rarely used.

Usually, the no. of radians is written like 2rad

You write degrees with a little circle at the top 1.2°

Same way, an angle of 1.2 radians would be written either as "1.2 rad" or "1.2 with a "c" at the top.(I can't seem to get the 'c' using formatting here.)

See-

http://en.wikipedia.org/wiki/Radian

, second paragraph last line for the answer to your question.(29 votes)

- Are negative degrees actual things, or are they hypothetical like negative numbers?(11 votes)
- they are actual things. For example, if you rotate an object 90 degrees clockwise, it would be -90 degrees. Like the number line, negative and positive only show direction(18 votes)

- Is 1.5 pi the same as 270?(8 votes)
- Yep, 1.5π radians is exactly 270°.

We usually use fractions for radians, so that would be 3π/2. What you said is completely correct, though!(12 votes)

- Is there any other way to measure the angle just like degrees, radians....?(6 votes)
- There was an attempt at a metric measure of angle where the right angle was divided into 100 parts (as opposed to the usual 90 degrees). The measure was called the gradian. There were 400 gradians in a complete revolution, and 1 gradian = 0.9 degrees.

It hasn't really caught on, and the only place I've seen it is on calculators.

Is that what you had in mind?

https://en.wikipedia.org/wiki/Gradian(12 votes)

- Why did humans invent radians and degrees? Isn't one enough?(5 votes)
- Radians make calculation easier in dealimg with derivatives.

https://qedinsight.wordpress.com/2011/03/14/why-radian-measure-makes-life-easier-in-mathematics-and-physics/

For example, if you take the derivative of sin x that will be

pi/180 cos x using degrees however by defining pi=180 the derivative will just be cos x which is simpler. You can get the result from the proof theorem on the derivative of sin x being cos x except instead of using radians as Sal does in his calculations use degrees. You will also notice that:

lim_x->0 sin x/x doesn`t equal 1 but pi/180 from using degrees by following the steps he carries out.

To understand the proof you should, however, have an understanding of limits/differentiation and circular geometry has in finding arc length and the area of a sector which you can learn about in some of Sal videos.(6 votes)

- When I have 5pi/9 and I am converting it to degrees do I set them up in the proportion just as in the video? And just like the example with -pi/2 do I multiply the 5 by 180? My answer was 100 degrees. Would like to know if I did it right. thx!!(5 votes)
- You are correct, 5π/9 * 180/π = 100. Notice you could multiply 5 * 180 and divide by 9 or you could divide 180/9 to get 20 and then multiply by 5 to get 100.(6 votes)

- why does Sal say, "radiussess"?(4 votes)
- There is a debate in math education that nobody needs to be told that "radiuses" is the plural of "radius", but "radii" has to be a vocabulary word. Since both words are acceptable plurals for "radius" in English, it would be clearer to teach with "radiuses", especially to students who don't speak English as their first language. I get the feeling in this video that Sal is new to this debate and is being a little silly about it. ^_^(4 votes)

- isn't -90 degrees 270 degrees?(4 votes)
- If the reference point is the positive x axis then yes, -90 degrees is 270 degrees.(4 votes)

## Video transcript

Lets see if we can give ourselves a little bit of practice converting between radians and degrees and degrees and radians. And just as a review lets just remind ourselves the relationship and I always do this before I have to convert between the two. If I do one revolution of a circle how many radians is that going to be? Well we know one revolution of a circle is (2)(pi) radians and how many degrees is that if I do one revolution around a circle? Well we know that is 360 degrees..... I can either write it with the little degree symbol like that or I can write it just like that. And this is really enough information for us to think about how to convert between radians and degrees if we want to simplify this a little bit we can divide both sides by 2. And you can have (pi) radians are equal to 180 degrees or another way to think about it going halfway around a circle in radians is (pi) radians. The arc that subtends that angle is (pi) radiusssses and that's also 180 degrees. And if you wanna really think about, well how many degrees are there per radian you can divide both sides of this by (pi). So if you divide both sides of this by (pi), you would get one radian it would have to go from plural to singular one radian is equal to 180/pi degrees. So all I did is I divided both sides by (pi) and if you wanted to figure out how many radians are there per degree you can divide both sides by 180. So you would get pi/180 radians is equal to 1 degree. So this is....now I think we are ready to start converting. So lets convert 30 degrees....to radians. So lets think about it. I'm going to write it out and actually this might remind you of unit analysis that you might do when you first did unit conversion but it also works out here. So if I were to write 30 degrees and this is how my brain likes to work with it I like to write out the word degrees. Well I wanna convert to radians so I really want to figure out how many radians are there per degree? So let me write this down. I wanna figure out how many radians do we have per degree? And I haven't filled out how many that is but we see the units will cancel out if we have degrees times radians/degree the degrees will cancel out and I'll be just left with radians. If I multiply the number of degrees I have times the number of radians per degree we're gonna get radians. And hopefully that makes intuitive sense as well. And here we just have to think about well if I have...think of it this way, if I have pi radians how many degrees is that? Well that's 180 degrees, comes straight out of this right over here pi radians for every 180 degrees or pi/180 radians/degree. This is going to get us to...we're going to get 30 times pi/180 30 times pi/180 which will simplify to 30/180 is 1/6 so this is equal to pi/6 let me write the units out this is 30 radians which is equal to pi/6 radians. Now lets go the other way lets think about if we have pi/3 radians and I wanna convert that to degrees. So what am I going to get if I convert that to degrees? Well here we're gonna want to figure out how many degrees are there per radian? One way to think about it is think about the pi and the 180 for every 180 degrees you have pi radians. 180 degrees/pi radians these are essentially the equivalent thing essentially you're just multiplying this quantity by 1 but you're changing the units the radians cancel out and then the pi's cancel out and you're left with 180/3 degrees 180/3 is 60 and we can either write out the word degrees or you can write degrees just like that. Now lets think about 45 degrees. So what about 45 degrees? And I'll write it like that just so you can figure it out with that notation as well. How many radians will this be equal to? Well once again we're gonna want to think about how many radians do we have per degree? So we're going to multiply this times well we know we have pi radians for every 180 degrees or we can even write it this way pi radians for every 180 degrees. And here this might be a little less intuitive the degrees cancel out and that's why I usually like to write out the word and you're left with 45 pi/180 radians. Actually let me write this with the words written out for me that's more intuitive when I'm thinking about it in terms of using the notation. So 45 degrees times, we have pi radians for every 180 degrees. So we are left with when you multiply 45 times pi over 180 the degrees have canceled out and you're just left with radians. Which is equal to what? 45 is half of 90 which is half of 180 so this is 1/4 this is equal to pi over 4 radians. Lets do 1 more over here. So lets say that we had negative pi/2 radians. What's that going to be in degrees? Well once again we have to figure out how many degrees are each of these radians. We know that there 180 degrees for every pi radians so we're gonna get the radians cancel out the pi's cancel out and so you have -180/2 this is -90 degrees or we can write it as -90 degrees. Anyway hopefully you found that helpful and I'll do a couple more example problems here because the more example for this the better and hopefully it will become a little bit intuitive