Radians and Degrees (NancyPi)

Published on May 16, 2018 by

MIT grad shows how to convert from radians to degrees, convert from degrees to radians, and explains what radians and degrees mean. To skip ahead: 1) For converting from DEGREES TO RADIANS, skip to time 4:50. 2) For converting from RADIANS TO DEGREES, skip to time 8:11. 3) For CONVERTING A DECIMAL number of radians into degrees, skip to time 10:50. Nancy formerly of MathBFF explains the steps.

For help with basic TRIGONOMETRY (SIN, COS, TAN), jump to: https://youtu.be/bSM7RNSbWhM

Follow me on Instagram! https://instagram.com/mathnancy
Follow me on Twitter: https://twitter.com/nancypi

Radians and degrees are just two different ways of expressing an angle, and they have different units. Angle measures written in degrees have a little superscript circle symbol after the number. Angle measures written in radians do not have the degree circle symbol and may or may not have the units "radians" or the abbreviation "rad." listed after the number.

1) TO CONVERT FROM DEGREES TO RADIANS: Multiply the number of degrees by the conversion factor PI OVER 180. For instance, to convert from 200 degrees into radians, you would multiply 200 degrees by the fraction quantity pi radians over 180 degrees. Since you are starting with degrees, and your conversion factor has degrees in the denominator, the degrees units will go away, or effectively cancel, and you will be left with the radian units you want. 200 times pi, divided by 180, simplifies to the quantity 10 pi over 9, which is now in radians. You can leave the symbol pi in your final answer.

2) TO CONVERT FROM RADIANS TO DEGREES: Multiply the number of radians by the conversion factor 180 OVER PI. For instance, to convert from 2 pi over 9 radians into degrees, you would multiply 2 pi over 9 by the fraction quantity 180 degrees over pi radians. Since you are starting with degrees, and your conversion factor has radians in the denominator, the radians units go away, or cancel out, and you will be left with the degree units you want. Since you are also starting with a pi in your angle measure, and the conversion factor has a pi in the denominator, the pi symbols will cancel out and you're left with just 2 times 180 over 9. 2 times 180 is 360, and 360 divided by 9 simplifies to 40, so your final answer is 40 degrees.

What is a radian? Technically, the definition is that if you had a circle with a radius r, an arc length distance of r around the circle would create an inside, central angle that is one radian.

How many degrees are in a circle? There are 360 degrees total in one full circle. Starting from zero degrees at the right-most point on the circle and moving counter-clockwise, one quarter of the way through a circle is 90 degrees, one half of the circle is 180 degrees, three-quarters of the way is 270 degrees, and one full circle is 360 degrees and brings you all the way back to your starting 0 degree point.

How many radians are in a circle? There are 2 pi radians total in one full circle. Starting from zero radians at the right-most point on the circle moving counter-clockwise, one quarter of the way through a circle is pi over 2 radians, one half of the circle is pi radians, three-quarters of the way is 3 pi over 2 radians, and one full circle is 2 pi radians and brings you back to your starting 0 radian point.

Notice that 360 degrees is equivalent to, or the same as, 2 pi. Since 360 degrees equals 2 pi, you could divide both sides of this equation by 2 and find that 180 degrees equals pi. You will use this equivalence, the fact that 180 degrees equals pi, to convert from degrees to radians, or from radians to degrees.

For more of my math videos, check out: http://nancypi.com

Category Tag


Advertise here. Telegram, Signal - Call/message +1-868-308-4028.


  • Manoj Bhardwaj 5 years ago

    Please upload a vedio about partial fraction….. Upload toughest question… 😁😁

  • Amiin hassen 5 years ago

    Yeah Nancy is back. We miss you, but I am in calculus 3 now, but I still like watching you. Big✊

  • Master Rahul Vlogs 5 years ago


  • Mando M 5 years ago

    I wish I could have a private tutor session with you Nancy to listen to your sweet voice for hours, then I’ll die a happy man.

    • Eleazar Chinne 5 years ago

      Mando M don’t we all my guy, don’t we all…

  • Gamezilla00 5 years ago

    She’s back!!

  • amty 5 years ago

    Nancy is so fancy

    • Bryston Bower 4 years ago

      More like 🔥💃hot

  • Upfrontgameplay 5 years ago


  • Giorgi Paikidze 5 years ago

    Thanks, I like math and your videos.

  • احمد 5 years ago

    From Iraq. …… continue❤☕❤

    • NancyPi 5 years ago

      Thank you 💚

    • احمد 5 years ago

      NancyPi I can not believe how Nancy responds to me😍😍😅

    • MrTheTonypp 4 years ago

      @NancyPi are you writing backwards??

  • Hassan Malla 5 years ago

    wonderful keep going 👉👉👉👉

  • BobC 5 years ago

    Nancy π rad.

    We knew that.

  • Ty Hollowell 5 years ago

    Hey Nancy!

  • 42TakenHydra 5 years ago

    I don’t understand stand if you write backwards or edit it

    • Vidhi Patel 4 years ago

      MrTheTonypp technique whatever same thing, and even if it’s mirrored she’s obviously gonna talk right…She’s just going to be flipped

    • MrTheTonypp 4 years ago

      @Vidhi Patel you are still wrong that would mean she wrote backwards before putting on mirror effect.

    • Vidhi Patel 4 years ago

      MrTheTonypp ok u really don’t understand this lol, let me explain. She is writing on a glass normally so that she can see it, NOT BACKWARDS. So obviously we can’t see it correctly.WHICH IS WHY she puts the mirror effect while editing it so that it basically FLIPS IT. So to summarize, in reality she is sitting on the right and writing so that she can see it correctly while it’s impossible for us to read, but after she “mirrors” the video, we can see it properly.

    • MrTheTonypp 4 years ago

      @Vidhi Patel I am willimg to bet if you ask her is she is writing backwards she will say yes

    • Vidhi Patel 4 years ago

      MrTheTonypp Deal.

  • 42TakenHydra 5 years ago

    And thank you

  • orp0piru 5 years ago

    angle & sin & cos \
    0 & sqrt{0}/2 & sqrt{4}/2 \
    pi/6 & sqrt{1}/2 & sqrt{3}/2 \
    pi/4 & sqrt{2}/2 & sqrt{2}/2 \
    pi/3 & sqrt{3}/2 & sqrt{1}/2 \
    pi/2 & sqrt{4}/2 & sqrt{0}/2

  • Synapsis12 5 years ago

    I have a head scratcher for you: I want to purchase a try square and accuracy is typically described “.001″ per inch”. I’m interested in one described as “.06 degrees” square. What does that convert to in thousandths of an inch?

  • OrphanPaper 5 years ago

    hi , what kind of voodoo is this

    • Bryston Bower 4 years ago


  • Deus Hamilton 5 years ago

    Filter all other videos out, Nancy knows best!

  • Seth Roche 5 years ago

    Hi Nancy, thanks for the pure awesomeness of your videos. 🙂 Have you done anything on Polar Coordinates?

  • Adil Shaikh 5 years ago

    Can you not use that pink marker? Can’r see numbers properly.

  • Hayatullah Sayedi 5 years ago

    Well done. Keep shining. 🙂

  • Abel Lopez 5 years ago


  • Ankit Sharma7255972503 5 years ago

    Salla bakwas hai

  • Prabhat Kumar 4 years ago

    Tnx i like math and your videos

  • David varela 4 years ago

    Nancy you just became my new best friend this semester 🤗

  • stuart paterson 4 years ago

    I absolutely love this! I haven’t done any serious maths for years outside of a bit of finance but your chilled approach is very cool and easy to follow. No doubt you will do some videos where i pull the ripcord early having no idea what’s going on but so far, so good… Thank you so much!

  • JB 4 years ago

    Thank you!!!!

  • DaelyLeandro22 4 years ago

    This was really helpful please upload more topics on pre-cal and calculus!!!

  • Ghizal Ahmad 4 years ago

    Love this video! Really helped me out in understanding the conversion better. 🙂 I was wondering whenever you had the time if you could possibly do a video on arc length? Including how to find both arc length and the central degree? Either way, thanks for all the help! 🙂

  • hitesh suryawanshi 4 years ago

    Thanks nancy

  • Barkat Shah 4 years ago

    Very useful like ur teaching style

  • kamoliddin chinaliyev 4 years ago

    I’m in Calculus-1 please continue with trigs

  • Jozef Aerts Wiskunde 4 years ago

    Degrees are rational ( ex 180 ) , but radians are irrational ( because Pi is irrational ).
    So how can 180 equal to Pi ??? Because a rational number can never be an irrational number

    • PhysicsHelps 4 years ago

      It’s true that we tend to use rational numbers with degrees and irrational numbers with radians, but any real number can work for both ways of expressing angles. Also, this is a unit conversion. 180 doesn’t equal pi, but 180 degrees does equal pi radians.

  • Oscar Ordiano 4 years ago


  • sumit kharb 4 years ago

    Madam mukherjee nagar aake pdha do naa, aap sahi taiyari kra doge ssc ki….

  • Walnut Grove Owners 4 years ago

    306 degrees = 2pi? Looks like one pie to me. LOL

  • alen zack 4 years ago

    how you can write like this

  • sean appletree 4 years ago

    Very nice presentation but somehow I cannot concentrate very well on the concepts as one might expect Nancy being so presentable and me being a male…… your so lovely Nancy…do you do private lessons…chuckle I am so naughty.

  • Farah Laith 4 years ago

    From Iraq yr so beautiful 💓

  • ilqar Flaqmanli 4 years ago


  • Shiedalicious 4 years ago

    thank u my nigga

  • Eric Klassen 4 years ago

    You are the cutest math instructor I have ever had. Thanks

    • Tee Se 3 years ago

      your like 5 centuries old bro thats so creepy

  • Al-haseeb academy of sciences 4 years ago

    love you

  • Евгений Глущенко 4 years ago

    Oh, it’s so hard to concentrate while viewing this video. Nancy is so… beautiful… I’m sorry, but it’s true.

    • James Lindgaard 3 years ago

      Да, ты правда. 😉

    • Marc Paradis 3 years ago

      Same here 🙂

  • Andy Jary 4 years ago

    Thank you for this!

  • Damien Kimball 4 years ago

    You need to set up a school. Your explanations are so clear and understandable
    Thank you

  • pts mech94 4 years ago

    put a video about the fourier series

  • jack comfirt 4 years ago


  • Egan Early 4 years ago

    is she writing backwards???!

  • Caleb Powell 4 years ago

    Is she writing backwards?

    • Yngvild Víðarr 4 years ago

      She probably flips it because you can do that on a computer.

  • Shakir MehXood 4 years ago

    TnX learned a lot

  • Shakir MehXood 4 years ago


  • Control Man 4 years ago

    How the heck does she write/draw backwards ????
    Rare to find someone so beautiful and so smart….Awesome Lady !!

    • Eta Carinae 4 years ago

      Matt Miller N8QZH I think she’s either left handed or the video is flipped

  • Camden Six 4 years ago

    1000th like is mine ahaha

  • Good Wind Maritime Services 4 years ago

    Nancy. Love your videos. Great review for me as I have not done any of this in 20 years. Thank you for the great explanation.

  • Abdinasir Yusuf 4 years ago

    To be honest, I have no idea what you’re saying, but I appreciate your hard work!

  • KK Gill 4 years ago

    My teacher just showed this in class on Friday and people were complaining bcuz a pretty girl wuz doing “nerdy videos”. If they think pretty girls can’t b smart I think they’re the ones with the problem. What im saying is I like ur videos; keep doing what ur doing. I like the message u r conveying. P.S U r very beautiful.

  • Barry Manilowa 4 years ago

    Wow! Alanis Morsette knows a lot about math!

  • Monty Py 4 years ago

    it’s do cool. she is writing in reverse direction.
    wait a second is it the flipped image of the real one.
    but she is beautiful. she is braking my attention.

    • jackson riviera 4 years ago

      lol flipped image, you’re right

  • Sultan Abdulla 4 years ago


  • Brendan Wilson 4 years ago

    Hi Nancy, thank you for your videos. Suggestion: can you do a video on why you can’t interchange degrees and radians in trigonometric arguments.


  • ELSolareli 4 years ago

    Thanks for the help! Nancy you are to sweet

  • Kittoo kuriosity 4 years ago

    It’s like u writing on my phone screen

  • Mohammad Abu Sehma 4 years ago

    Please mam read one time meaning of quran

  • Mohammad Abu Sehma 4 years ago

    Im saying this to you becoz i think it is better than billions like ,subscribe and comments !

  • Mohammad Abu Sehma 4 years ago

    Please respond

  • Craig Richardson 4 years ago

    Do a video on how to measure angles pls

  • Aryan Jain 4 years ago

    Hi sexy

  • Luis Torres 4 years ago


  • Mustapha Sammer 4 years ago

    Hi Nancy your classes are very interesting. I have almost 50 video the sum of my eyes and abbonnés is not even worth 2% dun of your video,! maybe they will see your sexy voice or your beautiful eyes in my chain, Maybe the numbers of my abbonates will increase what do you think ..? I am also a matematician and physisian.😍

  • Dale Putnam 4 years ago

    If you made a class on Udemy I would purchase it, and I have already purchased Krista’s

  • Rocky Kaim 4 years ago

    You awsome teacher, doing nice work for students ,I remembered my school time after this video.

  • Mathematics only 3 years ago

    Let’s suppose a circle of 2cm radius and an arc of length 2cm then the angle form by arc is the ratio of arc length / radius
    So 2cm/2cm = 1 (it is real no.

  • Karen Macpherson Wessinger 3 years ago

    thank you! awesome way of explaining this

  • Joe Smith 3 years ago

    Nice explanation 👌. That’s how it should be for every single matter. In fact for me, is more important the explanation rather than the operation itself. Just to make sure if u r doing right. explanation is the most important thing and u did it perfectly. More pls.

  • BleuKnight 3 years ago

    and she’s writing backwards

    • Adithya Ps 3 years ago

      It’s mirror mode

  • Center Minded 3 years ago

    Is she a teacher IRL?

  • Hector Heath 3 years ago

    If you think this through yourself (which is what you should do to understand it) this is easy stuff, just an exercise in proportions. Fabulous teaching though, Nancy.

  • Justin Lee 3 years ago

    I see that the calculator has three modes for using trig. and inverse trig. functions, which are degrees (deg), radians (rad), and gradians (grad). I know that we don’t talk about gradians in math class. I know that there is a conversion between degrees and radians with gradians.

  • Puchica Erika 3 years ago

    nancy come back and make videos

  • Tariq Maciver 3 years ago

    Bro I love your videos so freaking simple and straight to the point thank you and God Bless 🙏🏿

  • mario arrizon 3 years ago

    Thank you Nancy, you are a great teacher.

  • Malak Khalifa 3 years ago

    You are a great teacher i can understand every word you said although iam from arab country i can understand everything from you

  • Madhavi R 3 years ago

    Can you make more videos on calculus ( like integration)

  • John Brim 3 years ago

    I keep waiting for Nancy to write backwards.

  • Godfrey Katongo 3 years ago


  • Divya Viswanathan 3 years ago

    Wow your good

  • Melia Noshadian 3 years ago

    Thank you so much… your explanation is soooo cleaarrrrrr.

  • Jatin Saini 3 years ago

    Nice work
    Iss topic pr Mene bhi ek video banai h ek bar jarur dekho friends.

  • Mark Knight 3 years ago

    The spacing on this will probably be wrong and it not look like its supposed to but I learned it with this:
    Degrees x (pi symbol)
    (pi symbol) x radians

  • Alex Mortiis 3 years ago

    how can you write with your left hand and mirrored in the same time?

  • bob 3 years ago

    Hi Nancy Pi. I’m about 30 years late to watch these videos for school, but I am using them to brush up on math for programming. It would be beneficial for your viewers if you added worksheets in the narrative section (“SHOW MORE/SHOW LESS”) for practice to improve memory retention. Thank you for your work and videos.

  • grigaale 3 years ago

    my favorite

  • Christian Rodríguez Figueredo 3 years ago

    1.75 speed worked for me

  • HSB Purgatorials 3 years ago

    I wish I had an awesome math teacher like you in my school……..Thanks a bunch ma’am❤👏😀

  • jDUB1 3 years ago


  • Alex Xu 3 years ago

    at 4:06 lol i thought it said 11/180 i had to rewatch it twice XD

    • Edison Construction, LLC 3 years ago


  • Mr Joelly 3 years ago

    Is she Being held hostage?

  • terence wright 3 years ago

    Very good, excellent video.

  • blitzblutz 3 years ago

    I wish that I had these videos when I was in high school! Nancy is the greatest math teacher!

  • teja sam 3 years ago

    its radiance of nancy

  • mike henry 3 years ago

    Hay Nancy love your content…..question…..the last one wouldn’t 1.5 pi be 270 degrees…I was totally with you all the way up to therw5…please please please clarify for me peace out

  • JMDinOKC 3 years ago

    Sometimes she smiles a smile that used to be described as a “come-hither” smile; at least that’s how it’s interpreted by hormone-addled male college students. Even at MIT.

    • Chris Wright 3 years ago

      She’s right-handed. She then writes on the screen in the normal way and flips the video. The text is rendered the right way round and she appears left-handed.

  • Eurotropia 3 years ago

    Your voice makes me sleepy, your view makes me awake and your thoughts makes me amazed….so far so good! 🙂

  • Th1cc J3ff 3 years ago

    I’ve been banging my head on my desk trying to figure out why i keep getting the wrong answer, but… AS SOON AS NANCY SAID “DIVIDE BY PI”, I FLIPPED OUT. I FLIPPED OUT!!!! THANK YOU SO MUCH!! AHHHHHHHHHHHHHHHHHHHHHHH

    • Raspberry B 3 years ago

      Th1cc J3ff simp

    • Aurum Aurum 3 years ago

      @Raspberry B tf

    • Varsha 3 years ago

      @Raspberry B sure dude. stop begging for attention lmao

    • Raspberry B 3 years ago

      Varsha cringe

  • Keith Wilhelm 3 years ago

    I enjoy learning math with you.

  • Louis Chen 3 years ago


  • Ginny Franklin 3 years ago

    your so pretty Nancy

  • Charles Musama 3 years ago

    Nancy is a great teacher!

  • OM Pandey 3 years ago

    Hey Nancy I’m from India and I am too much satisfied by your videos thnq

  • Solomon DeMello 3 years ago

    Check your math Jesus doesn’t agree

  • Chris Wright 3 years ago

    I wish someone looked at me the way she looks at 2pi/9.

  • Chris Wright 3 years ago

    P.S. The writing backwards thing: it’s not that difficult. Once, after visiting a car showroom and having the salesman write upside-down so we could read what he was writing immediately across the desk, I taught myself to do this (for limited characters) and sometimes use it with students. However, my guess is that she’s right-handed, writes on the screen in normal orientation, then flips the video. The text then looks right from our perspective, and she appears left-handed.

  • Farme yar yar 3 years ago

    Yes nancyPi thanks for teaching

  • Adrian Charboneau 3 years ago

    Your an MIT grad? You look so uconfident but good writing baackwards skills

  • Felipe Ribeiro 3 years ago

    Between paying attention and telling my heart not to fall in love… so beautiful

    • Sajila Vijay 3 years ago


  • Aerdna YC 3 years ago

    You are the best!!! thank you so much!!

  • Luis Diaz 3 years ago

    Instead of dividing by pi….At 13:25 just multiply 1.5 times 57.3=85.95 in other words…(1.5 times (1 radian in degree = about 57.3)) that will get your answer.

  • Renu Sharma 3 years ago

    Can you write a little faster? 🥴🥴🥴🥴

  • EFE SHENEM 3 years ago

    hiiii nancy.

    why a circled is divided into 360 parts? for instance why we don’t divide it into 400 parts?

  • Math Ranger 3 years ago

    Learned a lot!

  • John Lark 3 years ago

    Wow …..you are amazing

  • Fuga Mante 3 years ago

    Lefties are way too powerful.😅

  • Tony Krebs 3 years ago

    Love your lessons! But how do you wright on the glass? Magic?

  • matar mbe 3 years ago

    I’d like to know each of the 46 wankers who disliked this video

  • charles golden 3 years ago

    amazing, have been looking for some time for proper explanation, didn’t know how to phrase a proper question. you have cleared for me so much. By the way you are very pretty.

  • old english clocks 1640 to 1700 3 years ago

    Damn I wish I had her as my math teacher in high school. I’m not stupid after all!

  • John Doe 3 years ago

    Back in the day, in the University I attend(ed) to, they wondered “How can we attract (have ’em enroll into) more students to the exact science – physics, math, statistics – programs?” I have the answer: By having you as student/teacher/asst teacher, the ammount of students would raise exponentially (at least male students). I mean, even if they couldn’t care less about science at all, neither they understand a word in the topic of each class/lecture, classes would be packed; they would attend every single class just to see/talk to you. The only calculus they would care about, is the odds (probability) of dating you. Maybe girls (the few of them in the programs) would feel a little intimidated by not being the center of atention anymore.😆 With all due respect the comment is both true and at the same time a recognition/compliment/tribute to your beauty and intelligence.

  • Aaron Ramos 3 years ago

    future wife goals.

  • Marco Ramonet 3 years ago

    “And you still have the pie, don’t forget the pie”

  • Grace PJS 3 years ago

    wow……..you’re amazing, well explained, i wish my lecturer was like you

  • tjiseclipsed 3 years ago

    Thanks my friend; you’ve helped explain these concepts in a much more clearly articulated manner than I’ve found in my texts so far.

  • Charles Abernathy 3 years ago


  • Dimitar Georgiev 3 years ago

    Very useful and well explained! Thank you! I am a big fan of you.

  • Angeles Morales 3 years ago

    what’s a good trick if you don’t have a zero on the top and bottom of the fraction when trying to solve in simplest form?

  • Niko 3 years ago

    I lovvvvvvvvvvvvvvvve your videos nancy you made me know more I am not a good speller but good in meth

  • Mustafa Asaad 3 years ago

    دكري تعرفين منو دكري😂😂

    دكري هو سيد علي 😂

  • mario marquez 3 years ago

    Hi Nancy

  • Mike Jouhari 3 years ago

    Hard to not fall in love with her!

  • GOURAB NATH 3 years ago


  • Cute Beauty 3 years ago

    thank you so much live saver 🙂

Comments are closed.