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### Course: Geometry (all content) > Unit 14

Lesson 4: Introduction to radians# Degrees to radians

To convert from degrees to radians, multiply the number of degrees by π/180. This will give you the measurement in radians. If you have an angle that's 90 degrees, and you want to know what it is in radians, you multiply 90 by π/180. This gives you π/2. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- But why do we keep the pi? can't we do (150*pi)/180=2.6 radians ? or do you need to keep the pi?(49 votes)
- A radian is a relative unit based on the circumference of a circle. As you know, radians are written as a fraction with a π, such as 2π/3, 5π/4, or 3π/2. Since the equation for the circumference of a circle is C=2πr, we have to keep the π to show that it is a portion of the circle. Radian values can be used to calculate arc length using the radian and the radius multiplied together. Since it is encouraged to write these lengths in π units, The symbol is left give a π radian value.(46 votes)

- At1:25, Sal wrote that the circumference is 2pi radii, then at1:54, Sal wrote 2pi radians:

Is a radian equal to the radius. Or is it the same thing, just named differently? I mean, isn't the radian basically the radius bent around on the circumference. I'm really confused right now, so I would really appreciate it if someone would clearly explain this to me... the simpler the explanation, the better :) Thanks in advance!(19 votes)- You're right about the way you visualize the definition of a radian. It is "the angle subtended by an arc equal in length to the radius." That is, one radian is the
*angle*you would go through if you went one radius-worth of length on the circle. The radius, however, is a length measurement while the radian is an angle measurement.(38 votes)

- On my calculator , I have three angle notations, DRG. I know that D and R are degrees and radians respectively, so I checked on my calculator what it was. It turns out that it was gradians. So I looked it up and realized that a gradian is 10/9 a degree but I couldn't understand what it is used for. Could someone please answer my question? Thanks!(13 votes)
- A gradian is 1/100th of a right angle. It was introduced when France was trying to make everything metric (they had a metric calendar too: it didn't work well). So they took the right angle, the "most natural" angle, and divided it into a hundred parts. So there's 400 gradians to a circle.

So 400 grad=360° Divide by 360 and get 10/9 grad=1°(26 votes)

- What does a question mean when is says to convert Degrees to Radians in terms of Pi?(7 votes)
- That means not to estimate the value of π, but to leave your answer containing the symbol for π.

For example, 90° in radians is ½π.(21 votes)

- Is 2 pi radians equal to 2 pi radius?(7 votes)
- They are different.

Radians is a unit of measure like degrees. 2 pi radians, means you have 2 pi of something and they are radians. It is the same 360 degrees.

A radius is the line from the side of the circle to the center. Sometimes radius refers to the lengh of that line. 2 pi radius means multiply 2 pi by the length of the radius which will give you half of the circumference. It is a length that changes depending on the size of the circle.(14 votes)

- Is there a symbol for radians?(7 votes)
- No. Radians is the ratio the the arc length to the radius, thus the units cancel out, thus there are no units. You can write in "radians" if you like, but it is not required.(14 votes)

- Radiuseseseseseseseses.(10 votes)
- Yep. Nice joke by Sal. But do post in the tips and thanks(5 votes)

- How would you go about changing degrees to radians when you do not have a whole number? (I have a problem that's asking for 97.5 deg)

I see the answer is 13/24 pi

but the pi is also after the fraction.

Is this a completely different method?(6 votes)- No. The method is the same. You would multiply by pi/180

97.5 * pi/180 = 97.5 pi/180

You do need to clean up the fraction as you should have two integers. So, muliplty by 2/2 to get? 195pi/360

Last, reduce the fraction by 15: 13pi/24 which is the same as (13/24) pi(9 votes)

- How can it be that 45°=45pi/180 radians while sin(45°)=sqr(2)/2? Can anyone help me to bring those two things together?(4 votes)
- Oh, I think I just made up my mind about this: 45°=45pi/180 radians is the stretch of the circle at 45° and sin(45°)=sqr(2)/2 the y-value (or opposite side of triangle) at 45°.(8 votes)

- Why exactly do we need to learn radians? Do they use it countries other than America? I have never heard someone measure something in radians-.(5 votes)
- Radians are a more natural way to measure angles. Why are there 360 degrees in a circle? The number 360 has high divisibility but is still pretty arbitrary. Why isn't a full circle defined to be 100 degrees or something else instead?

Radians, on the other hand, depend on the geometry of a circle rather than an arbitrary number. This is why radians are more natural. As a result, in higher-level math, you will almost always use radians instead of degrees.

You also asked if other countries besides America use radians. Other countries use degrees but they also use radians too. In fact, if we were ever to meet an alien civilization, they would measure angles using radians (but not necessarily degrees). Radians are a "universal" angle measure.(5 votes)

## Video transcript

- [Instructor] We're asked
to convert 150 degrees and negative 45 degrees to radians. Let's think about the relationship between degrees and
radians, and to do that, let me just draw a little circle here. So that's the center of the circle, and then do my best shot, best attempt to freehand draw a
reasonable-looking circle. That's not, I've done worse than that. Alright, now, if we were to go in degrees, if we were to go one
time around the circle like that, how many degrees is that? We know that that would be 360 degrees. If we did the same thing,
how many radians is that, if we were to go all the
way around the circle? We just have to remember,
when we're measuring in terms of radians, we're really talking about the arc that subtends that angle. So if you go all the way around, you're really talking about the arc length of the entire circle, or essentially the circumference of the circle. And you're essentially
saying, how many radius's this is, or radii, or how many radii is the circumference of the circle. You know a circumference
of a circle is two pi times the radius, or you could say that the length of the circumference of the circle is two pi radii. If you wanna know the
exact length, you just have to get the length of the radius
and multiply it by two pi. That just comes from the, really, actually the definition of pi, but
it comes from what we know as the formula for the
circumference of a circle. If we were to go all the way around this, this is also two pi radians. That tells us that two pi radians, as an angle measure, is
the exact same thing, and I'm gonna write it out, as 360 360 degrees. And then we can take
all of this relationship and manipulate it in different ways. If we wanna simplify a little bit, we can divide both sides
of this equation by two, in which case, you are left with, if you divide both sides by two, you are left with pi radians is equal to 180 degrees. How can we use this relationship now to figure out what 150 degrees is? Well, this relationship, we could write it in different ways. We could divide both sides by 180 degrees, and we could get pi radians over 180 degrees is equal to one, which is just another way of saying that there are pi radians for every 180 degrees, or you could say, pi over 180 radians per degree. The other option, you could divide both sides of this by pi radians. You could say, you would
get on the left hand side you'd get one, and you would also get, on the right hand side,
you would get 180 degrees for every pi radians. Or you could interpret this as 180 over pi degrees per radian. How would we figure out, how
would we do what they asked us? Let's convert 150 degrees to radians. Let me write the word out. So, 150 degrees. Well, we wanna convert this to radians, so we really care about
how many radians there are per degree, actually, let
me do that in that color. (typing) We care about how many radians there are per degree. We'll do that same green color. Per degree. How many radians are there per degree? Well, we already know, there's pi radians for every 180 degrees, or there are pi... Let me do that yellow color. There are pi over 180 radians per degree. And so, if we multiply,
and this all works out because you have degrees in the numerator, degrees in the denominator,
these cancel out, and so you are left with 150 times pi divided by 180 radians. So what do we get? This becomes, let me just rewrite it. 150 times pi. All of that over 180, so this is equal to, and
we get it in radians. And so, if we simplify it, let's see, we can divide the numerator
and the denominator both by, looks like, 30. So if you divide the
numerator by 30, you get five. You divide the denominator
by 30, you get six. So you get five pi over six radians, or 5/6 pi radians, depending
how you wanna do it. Now let's do the same thing
for negative 45 degrees. What do you get for negative 45 degrees if you were to convert that to radians? Same exact process. You have negative, and I'll
do this one a little quicker. Negative 45 degrees. I'll write down the word. Times, times pi radians, pi radians for every 180 degrees. The degrees cancel out, and you're left with negative 45 pi over 180 radians. So this is equal to negative 45 pi over 180, over 180 radians. How can we simplify this? Well it looks like
they're both, at minimum, divisible by nine, nine times five is 45, this is nine times 20,
so actually it's gonna be divisible by more than just, let's see... Actually, they're both divisible by 45. What am I doing? If you divide the numerator
by 45, you get one. You divide the denominator by 45, 45 goes into 180 four times. You're left with negative
pi over four radians. This is equal to negative pi over four radians. And we are done.