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## High school geometry

# Radians to degrees

CCSS.Math:

To change from radians to degrees, you need to multiply the number of radians by 180/π. This number will help you switch between the two units. For example, if you multiply π/2 radians by 180/π, you will get 90 degrees. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What if the radians come in decimals. For example I came across to having to convert 0.9 radians into degrees. How would you be able to convert that when it's not in terms of pi ?(22 votes)
- For any amount of radians, whole or decimal, positive or negative, you just multiply by [180/pi] to get degrees. So 0.9 * 180 / pi ~=
**51.57 deg**.(56 votes)

- I like this website but these don't explain why I can't use the value for one radian and multiply that by the degrees I've been given to convert? Why does 240 degrees get converted into 4π over 3 radians? shouldn't it be 13750.99 lol? Or am I simply too stupid to understand this concept?(9 votes)
- We usually use the fractions with pi when talking about radians because it is actually easier to work with the fractions. (I know, a lot of people don't like fractions, but they are our friends!) It is easier to work with 4/3 pi than the decimal equivalent which is 4.188790. (I think you used the wrong conversion process to get your 13750.99. That is an upside down conversion that you would get if you multiplied 240 times 180 and then divided by pi. Rather than giving you radians, it gave you degrees squared :) Sal shows us how to line up the units so they cancel in this video and his video on converting degrees to radians, which is just the inverse of radians degrees--also in the units videos on Khan Academy.)

The size of the angle is exact when you use the fraction, but when you convert to decimals, most of your results are NOT exact--they are approximations. They get very messy when you do the next step, and the**next**step with your results. In more advanced math, your first results are just stepping stones for all the other steps you need to do, so messy is not good.

Another reason is that some angles show up in problems over and over, so they become old friends even when they are something like 4/3 pi. The diagonal of a square forms a 45 degree angle which is pi/4. Half of an equilateral triangle forms a 30-60-90 degree triangle. 30 degrees is pi/6 and Sal just showed us that 60 degrees is pi/3. A right angle is 90 degrees and that is pi/2.

You will get a chance to work with plenty of decimals, though, because if a number of degrees does not form a nice fraction with 180, then we convert all the way to a decimal and just have to deal with the decimal places.(42 votes)

- At0:21, Sal says that 2pi radians is 360 degrees. When you do basic geometry, 2pi radius (radii) is 360 degrees. As both statement are equal, are radius's and radians the same or different? Please give the definition of both terms.(6 votes)
- They're different.

Radius: the distance between the center of a circle and any point on its circumference

Radian: One radian is 180/pi degrees. An arc of a circle (that has a radius of 1 unit) with a central angle of a radians has a length of a units.(14 votes)

- I don't understand how to convert decimal degree measures into degrees-minutes-seconds, can anyone help?(7 votes)
- To convert 15.6358 degrees to deg min sec form do:

Clearly we have 15 degrees, so the remaining 0.6358 is minutes and seconds.

Since one degree has 60 min, we can write that x = 0.6358 * 60 So x is 38.148 min.

A simpler example would be 0.5 degrees is equivalent to 0.5 * 60 = 30 minutes, so half of one degree, which makes sense.

Now we convert the 0.148 remaining min to sec in a similar manner.

y = 0.148 * 60 so y is 8.88 sec. Since there isn't really something smaller then a sec, we leave it at that.

The final result is 15 deg 38 min 8.88 sec(16 votes)

- what is meant by radians(6 votes)
- radians are just another form of measurements that can be used to scale things with larger form. anything measured in degrees can also be measured in radians. if we are working on a question with the degrees of a circle we could go about it as 360degrees or we could work the problem as 180radians. now if we were working with triangle using degrees would prob be a bit more useful.hope this helped(10 votes)

- Why doesn't radians have a symbol. And if it does, what is it?(5 votes)
- Radians can be represented by a
**superscript "c" symbol after the angle measure in radians**.

The "c" here stands for circular measure.

However this symbol is rarely used as it can be easily confused with the degree symbol(°).

Instead mostly the words**radians or "rads" is written after a measure in radians**if clarity is required.(8 votes)

- could I also get help for this problem? 17pi/18 rads to degrees. the way the video describes it doesn't explain for this.(2 votes)
- (17π radians) / 18

To convert from radians to degrees, multiply the angle measure by 180º / (π radians):

(17π radians)/ 18

(17π radians)(180º) / 18(π radians)

17(180º) / 18*170º*

That's the answer!(8 votes)

- How can something be a 'negative' radian (-pi/3)? How can a radian be negative?(3 votes)
- The convention is to take counter clockwise as positive. So if you go clockwise π/3 radians from the 0 radian position, then your angle measure is -π/3 radians.(4 votes)

- I don't really understand this, since the example given was with pi alone, but what about a number like 23pi/20?(3 votes)
- Your number is in radians. To convert from radians to degrees, you multiply by 180/pi.

Hope this helps.(4 votes)

- How can you remember all these steps?(3 votes)
- have a good memory(2 votes)

## Video transcript

We're asked to convert pi radians and negative pi/3 radians to degrees. And the first question I'll ask you: If you do one revolution, You have an angle that went all the way around once. How many radians is that? Well we know that it is 2 pi radians. Now that exact same angle if we were to measure it in degrees, How many degrees is that? Well if you were doing degrees, it would be one full revolution. That is equal to 360 degrees Now, can we simplify this? That's a bore to write this little, superscript circle That's literally the units of the question. Sometimes it doesn't look like a unit but it is a unit. You could literally write degrees instead of that little symbol. Now can we simplify this a little bit? Well sure, Both two pi and 360 are divisible by two so lets divide things by two, and if we do that, what do we get? Or what are pi radians equal to? Well on the left side here we're just left with pi radians, and on the righthand side here, 360 divided by two is 180. And we have still the units which are degrees. So we get pi radians are equal to 180 degrees. Which actually answers the first part of our question. We wanted to convert pi radians, well we just figured out! Pi radians are equal to 180 degrees. If you want to think about it, pi radians are halfway around the circle Halfway around the circle like that, and it is the same thing as 180 degrees. So now lets think about the second part. We want to convertnegative pi over three radians. --Switch to a new color-- so negative pi over three, so how do we convert that? So what do we get based on this information right over here. Well, to figure this out we need to know how many degrees there are per radian. We need to multiply this by degrees -- I'm going to write the word out instead of the circle here -- It would be really hard to visualize that, degrees per radian So how many degrees are there per radian? well we know that for 180 degrees we have pi radians. Or you can say there are 180 over pi degrees per radian. This is going to work out: We have however manyradians we have times the number of degrees per radian. So of course the units are going to work out. The radians cancel out, the pi also cancels. And you are left with 180 divided by 3, leaving us with what is that? Negative 60, and we don't want to forget the units We could write them out, the only unit left is degrees. WE could write out the word degrees or just put that symbol there.