Intuitive Guide to Angles, Degrees and Radians

It’s an obvious fact that circles should have 360 degrees. Right?

Wrong. Most of us have no idea why there’s 360 degrees in a circle. We memorize a magic number as the “size of a circle” and set ourselves up for confusion when studying advanced math or physics, with their so called “radians”.

“Radians make math easier!” the experts say, without a simple reason why (discussions involving Taylor series are not simple). Today we’ll uncover what radians really are, and the intuitive reason they make math easier.

Where Do Degrees Come From?

Before numbers and language we had the stars. Ancient civilizations used astronomy to mark the seasons, predict the future, and appease the gods (when making human sacrifices, they’d better be on time).

How is this relevant to angles? Well, bub, riddle me this: isn’t it strange that a circle has 360 degrees and a year has 365 days? And isn’t it weird that constellations just happen to circle the sky during the course of a year?

Unlike a pirate, I bet you landlubbers can’t determine the seasons by the night sky. Here’s the Big Dipper (Great Bear) as seen from New York City in 2008 (try any city):

constellation rotation

Constellations make a circle every day (video). If you look at the same time every day (midnight), they will also make a circle throughout the year. Here’s a theory about how degrees came to pass:

  • Humans noticed that constellations moved in a full circle every year
  • Every day, they were off by a tiny bit (“a degree”)
  • Since a year has about 360 days, a circle had 360 degrees

But, but… why not 365 degrees in a circle?

Cut ‘em some slack: they had sundials and didn’t know a year should have a convenient 365.242199 degrees like you do.

360 is close enough for government work. It fits nicely into the Babylonian base-60 number system, and divides well (by 2, 3, 4, 6, 10, 12, 15, 30, 45, 90… you get the idea).

Basing Mathematics on the Sun Seems Perfectly Reasonable

Earth lucked out: ~360 is a great number of days to have in a year. But it does seem arbitrary: on Mars we’d have roughly ~680 degrees in a circle, for the longer Martian year. And in parts of Europe they’ve used gradians, where you divide a circle into 400 pieces.

Many explanations stop here saying, “Well, the degree is arbitrary but we need to pick some number.” Not here: we’ll see that the entire premise of the degree is backwards.

Radians Rule, Degrees Drool

A degree is the amount I, an observer, need to tilt my head to see you, the mover. It’s a tad self-centered, don’t you think?

Suppose you saw a friend go running on a large track:

“Hey Bill, how far did you go?”

“Well, I had a really good pace, I think I went 6 or 7 mile–”

“Shuddup. How far did I turn my head to see you move?”


“I’ll use small words for you. Me in middle of track. You ran around. How…much…did…I…turn…my…head?”


Selfish, right? That’s how we do math! We write equations in terms of “Hey, how far did I turn my head see that planet/pendulum/wheel move?”. I bet you’ve never bothered to think about the pendulum’s feelings, hopes and dreams.

Do you think the equations of physics should be made simple for the mover or observer?

Radians: The Unselfish Choice

Much of physics (and life!) involves leaving your reference frame and seeing things from another’s viewpoint. Instead of wondering how far we tilted our heads, consider how far the other person moved.

Degrees vs radians

Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.

But absolute distance isn’t that useful, since going 10 miles is a different number of laps depending on the track. So we divide by radius to get a normalized angle:

\displaystyle{Radian = \frac{distance \hspace{10pt} traveled}{radius}}

You’ll often see this as

\displaystyle{\theta = \frac{s}{r}}

or angle in radians (theta) is arc length (s) divided by radius (r).

A circle has 360 degrees or 2pi radians — going all the way around is 2 * pi * r / r. So a radian is about 360 /(2 * pi) or 57.3 degrees.

Now don’t be like me, memorizing this thinking “Great, another unit. 57.3 degrees is so weird.” Because it is weird when you’re still thinking about you!

Moving 1 radian (unit) is a perfectly normal distance to travel. Put another way, our idea of a “clean, 90 degree angle” means the mover goes a very unclean pi/2 units. Think about it — “Hey Bill, can you run 90 degrees for me? What’s that? Oh, yeah, that’d be pi/2 miles from your point of view.” The strangeness goes both ways.

Radians are the empathetic way to do math — a shift from away from head tilting and towards the mover’s perspective.

What’s in a name?

Radians are a count of distance in terms of “radius units”, and I think of “radian” as shorthand for that concept.

Strictly speaking, radians are just a number like 1.5 or 73, and don’t have any units (in the calculation “radians = distance traveled / radius”, we see length is divided by length, so any units would cancel).

But practically speaking, we’re not math robots, and it helps to think of radians as “distance” traveled on a unit circle.

Using Radians

I’m still getting used to thinking in radians. But we encounter the concept of “mover’s distance” quite a bit:

  • We use “rotations per minute” not “degrees per second” when measuring certain rotational speeds. This is a shift towards the mover’s reference point (“How many laps has it gone?”) and away from an arbitrary degree measure.

  • When a satellite orbits the Earth, we understand its speed in “miles per hour”, not “degrees per hour”. Now divide by the distance to the satellite and you get the orbital speed in radians per hour.

  • Sine, that wonderful function, is defined in terms of radians as

\displaystyle{sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} ...}

This formula only works when x is in radians! Why? Well, sine is fundamentally related to distance moved, not head-tilting. But we’ll save that discussion for another day.

Radian Example 1: Wheels of the Bus

Let’s try a real example: you have a bus with wheels of radius 2 meters (it’s a monster truck bus). I’ll say how fast the wheels are turning and you say how fast the bus is moving. Ready?

“The wheels are turning 2000 degrees per second”. You’d think:

  • Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second. Circumference = 2 * pi * r, so it’s moving, um, 2 * 3.14 * 5 and 5/9ths… where’s my calculator

“The wheels are turning 6 radians per second”. You’d think:

  • Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.

Wow! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover.

The reverse is easy too. Suppose you’re cruising 90 feet per second on the highway (60 miles per hour) on your 24″ inch rims (radius 1 foot). How fast are the wheels turning?

Well, 90 feet per second / 1 foot radius = 90 radians per second.

That was easy. I suspect rappers sing about 24″ rims for this very reason.

Radian Example 2: sin(x)

Time for a beefier example. Calculus is about many things, and one is what happens when numbers get really big or really small.

Choose a number of degrees (x), and put sin(x) into your calculator:

When you make x small, like .01, sin(x) gets small as well. And the ratio of sin(x)/x seems to be about .017 — what does that mean? Even stranger, what does it mean to multiply or divide by a degree? Can you have square or cubic degrees?

Radians to the rescue! Knowing they refer to distance traveled (they’re not just a ratio!), we can interpret the equation this way:

  • x is how far you traveled along a circle
  • sin(x) is how high on the circle you are

So sin(x)/x is the ratio of how high you are to how far you’ve gone: the amount of energy that went in an “upward” direction. If you move vertically, that ratio is 100%. If you move horizontally, that ratio is 0%.

sin x vs x

When something moves a tiny amount, such as 0 to 1 degree from our perspective, it’s basically going straight up. If you go an even smaller amount, from 0 to .00001 degrees, it’s really going straight up. The distance traveled (x) is very close to the height (sin(x)).

As x shrinks, the ratio gets closer to 100% — more motion is straight up. Radians help us see, intuitively, why sin(x)/x approaches 1 as x gets tiny. We’re just nudging along a tiny amount in a vertical direction. By the way, this also explains why sin(x) ~ x for small numbers.

Sure, you can rigorously prove this using calculus, but the radian intuition helps you understand it.

Remember, these relationships only work when measuring angles with radians. With degrees, you’re comparing your height on a circle (sin(x)) with how far some observer tilted their head (x degrees), and it gets ugly fast.

So What’s the Point?

Degrees have their place: in our own lives, we’re the focal point and want to see how things affect us. How much do I tilt my telescope, spin my snowboard, or turn my steering wheel?

With natural laws, we’re an observer describing the motion of others. Radians are about them, not us. It took me many years to realize that:

  • Degrees are arbitrary because they’re based on the sun (365 days ~ 360 degrees), but they are backwards because they are from the observer’s perspective.
  • Because radians are in terms of the mover, equations “click into place”. Converting rotational to linear speed is easy, and ideas like sin(x)/x make sense.

Even angles can be seen from more than one viewpoint, and understanding radians makes math and physics equations more intuitive. Happy math.

Other Posts In This Series

  1. A Visual, Intuitive Guide to Imaginary Numbers
  2. Intuitive Guide to Angles, Degrees and Radians
  3. Intuitive Arithmetic With Complex Numbers
  4. Understanding Why Complex Multiplication Works
  5. Intuitive Understanding Of Euler's Formula
  6. An Interactive Guide To The Fourier Transform
  7. Intuitive Understanding of Sine Waves
  8. An Intuitive Guide to Linear Algebra

Questions & Contributions


  1. Hi Khalid, I always enjoy your explanations.

    While you’re talking about how arbitrary degrees are as a measure of angle, it reminded me of how radians are somewhat arbitrary. Here’s a really interesting article on how pi should actually be defined as what is currently 2 * pi:

  2. Hi Tony, thanks for the comment! Yep, that’s a very interesting article.

    I think radians are pretty natural (distance traveled/radius), but the scale is made somewhat arbitrary (as you say) based on our definition of pi. Perhaps it would be easier if a circle had pi radians instead of 2 pi (as a result of defining pi to be the circumference of a unit circle).

  3. Shouldn’t the orbital speed of the satellite be the linear speed in mph divided by the distance from the center of the earth to the satellite (not the radius of the earth)?

  4. I’m a pilot. I can just imagine ATC telling me “Turn right heading pi radians.” Maybe they would just give me the coefficient and say turn right heading 1. Then north could be zero, east could be .5 etc. I can also imagine looking at my compass in the plane and seeing it marked with 0, 1/2pi, pi, 1 1/2 pi, 2pi, etc. Actually…no, I can’t imagine any of that at all. :)

  5. @Tracy: Heh, point well taken — degrees are definitely best when we’re observing our own motion :). Though a radian compass might be a fun gag.

  6. Kalid I can only encourage you to write faster and write more. I can’t get enough of these explanations.

  7. HI KALID,
    i am an comp sc grad. working for 15 yrs.
    i love maths, but i had really bad maths teachers, all thru school and college.

    as a result, i never understood what radians were for. THANKS A TON for this article. at last it is crystal clear.


  8. Thanks av, glad it was helpful! It took me quite a while to figure out what radians were about as well :).

  9. Kalid,
    Thank you again! I agree with the others. More!! That is all long as the quality stays the same :-) and you have time to work out.


  10. Dear kalid,
    Your site has rekindled my interest in Maths. thanks a lot.
    Recently I had started assuming the following;
    Degrees: Angle measured from origin
    Radians: Angle measured from circumference (in terms of radius)
    For equilateral triangle, the angle is 60 degrees between two sides. If these two sides are squeezed to form 57.3 degrees, third side bulges out to form an arc of a circle with 1 radian measurement.


  11. @Mr. Rose: Thanks — yep, will definitely try to keep the quality up :).

    @V.Manoharan: Glad you enjoyed it. That’s an interesting thought — yes, an angle of 1 radian (about 57.3 degrees) will correspond to a bulge of length 1.

  12. hi khalid!

    i m a student of 11th std, and i really hated maths before meeting you, but you are a real eye opener!!

    P.S. i wish i had teachers like you in school!!!!

  13. (in continuation)…..
    though i have become a great fan of yours, i just want to say that you should try to cover up topics a bit faster as you are my teacher from now onwards…..i m from india, and prep. for engineering….hope you consider it(covering up topics)…especially trigo. and quadratic equations, they are my least favourite(very tough)
    thank you,

  14. Hi Chirag, thanks for the comment! Yep, math can be enjoyable if seen in the proper light.

    I’ll try to keep cranking out posts as I can :)

  15. Your explanation of constellation rotation is incorrect. Every constellation rotates completely everyday. So if the Big Dipper is upside down it’ll be right side up in twelve hours. Now if you measure at the same time everyday each constellation will be one degree farther along than at the same time the day before and that is the once a year rotation. Your wikipedia reference has a succint description. “Ancient astronomers noticed that the stars in the sky, which circle the celestial pole every day, seem to advance in that circle by approximately one-360th of a circle, i.e., one degree, each day.”

    Otherwise great post. Thanks.

  16. Obama needs to add you to the educational advisory board :-) Your method of teaching definitely networks more of the mind enabling better recall and retention. I am sure that when Benjamin Franklin was creating the core of the current educational policies of this nation, this was way closer to the mark of what he intended then the holes in the head that we currently have.

  17. I have a question that I hope you can answer. When talking about a 360 degree circle, 90 degrees is vertical and 180 degrees is horizontal. Why is that different in cardinal directions? Example, 0 degrees is North. I am trying to understand the difference.

    Thank you

    Kim Frey

  18. @Anon: Thanks :). Yes, I think there are many improvements we can make to how education is handled.

    @Kim: Great question. I think the difference is in the starting reference point.

    Mathematicians are used to thinking about the x and y axis, so going “right” on the x-axis is the natural starting point for them. Therefore +90 degrees means going “North” (or along the y-axis, as angles increase counter-clockwise for mathematicians).

    In navigation (like hiking in the woods), North may be a more universal reference point — it’s where compasses point. In that case, +90 degrees means going East (since the angles increase clockwise).

    It’s a bit confusing since each type of use has a different reference point, but thanks for asking.

  19. How do radians and degrees relate to sine space or sines? My understanding of sines is that you simply compute the ratio of the opposite/hypotenuse sides of a right triangle to derive the linear measurement. So can I infer that sines are also from the perspective of the “mover”; since you are dividing one length by another length? If so, would it be better to use sine space because you would have a dimension associated to your unit (cm, m, km, etc…). Maybe a discussion on how the right triangle relates to the unit circle would be helpful. Thanks for a great article!

  20. @E.G.: I’m not quite sure I understand the question, but here’s my take on how sine and radians relate.

    Radians and degrees represent progress along a circle; 90 degrees represents a quarter-turn, and pi/2 radians represents the distance traveled when moving a quarter of the way around the circle.

    Sine can mean many things, including the ratio of the sides of a right triangle. Another interpretation which may help is that sine represents the “height”, where 1.0 is the max height, -1.0 is the min height, and everything else is a fraction in-between.

    (Edit: correcting an error in this comment):

    The interesting thing is that sine/cosine represent position in grid coordinates, which the mover may not know about!

    For example, 45 degrees represents a certain position along the circle. From the mover’s perspective, they are halfway to 90 (top of the circle), and indeed, they have moved halfway to their goal (at 45 degrees, the distance along the circumference from the start and top of the circle is the same).

    But from our observer’s perspective, 45 degrees looks like a height of sin(45) or .707 — that is, at 45 degrees, the mover is 70.7% of the way to the top! In the last “half” they move the remaining amount. I see sine and cosine as ways for us to map the distance traveled in the mover’s frame of reference to distance traveled in ours.

    Radians and degrees are different ways of describing how far you’ve traveled along the circle. Sine is a way to describing how ‘high’ you are on the circle (from our grid’s perspective), as a percentage of the maximum. Hope this helps!

  21. Thanks for the explanation. My intent was to understand the topic from a different perspective. Now I understand (“a ha”), thanks!


  22. I finally got this radian stuff and thought I would sum up my brain blast for anyone who is still confused. This is a really basic explanation that just uses straight math.

    Describing an angle in radians is just a way of writing an angle without the degree symbol.

    The measure of an angle in radians is the ratio of the arc length it cuts out to the length of the circle’s radius.

    If the arc that the angle cuts out is exactly equal to the length of the radius, the angle therefore has a measure of 1/1, 1 radian, or just 1.

    Another useful example:
    If the angle cuts out an arc that is equal to the whole circumference of the circle (2πr), the normal angle is 360 degrees, and so that means that the ratio of the arc length (2πr) to the radius is just 2πr/r, or 2π. This gives us the measure of the angle in radians, 2π.

    In other words, 360 degrees = 2π radians,
    180 degrees = π radians, and
    180 degrees/π radians = 1.

    We can use this ratio as a conversion factor.

    To convert 39 degrees to radians, multiply 39 degrees x (π radians/180 degrees) to find that 39 degrees is really 39π/180 or about 0.68 radians!

    I hope that helped!

  23. hey… ur explanation is really good. i have a presentation on radian angle tomorrow… do u knoe where i can find some more funda on radian???

  24. Hi Kalid I’m in grade 11 doing a college course supplied by our school and right now I think radians are the most stupid “number” in the world. Thanks for the explanation on what the stupid things are any way at least I’ll be able to understand part of it.

  25. I see the diff between Radians and Degrees as, according to the above formula 1 Radian = 57.3 Degrees BUT if I use a scientific calculator, the ratio between a Sine(x) in Radians and a Sine(x) in Degrees is never 57.3 If I process in C# Math.sin(x), I get an answer in Radians, but mutiplying that answer by 57.3 gives a different outcome to that on a calculator. What am I missing?

  26. @George: Thanks for the comment! Yep, you can see radians as a ratio between the distance traveled on the circle and the radius.

    @azariah007: Glad you’re able to find it useful –after a while you might see radians starting to make sense :).

    @Dave: Great question. Radians -> degree conversions need to happen when you’re inputting the angle; after you’ve run sin, the result will be vastly different.

    So, you can say

    sin(d) = sin(r)

    where d = 57.3 * r. So, you could pick r = 1 (and d = 57.3) and get the same result.

  27. Math is soooooooooooooooo interesting to me with the numbers and the signs and the stuff that makes numbers make sense. Ahhhhh amazinggggg!!!!!!!!! (SMILING REAL BIGG)

  28. You could say that I am a bit lazy when it comes to posting comments – but to miss a chance to say how great this is would be too much to lose

  29. Thank you for your explanation, it made a lot more sense than the others I’ve seen (which basically just say “It’s a radian. Get over it.” but in a posher way.)

    The only thing which is still niggling away at me is that 57 degrees does not divide into 360 degrees neatly. I mean, one foot is 12 inches, not 11.67 inches. I find that ugly, but I guess it is all part of the wonderful mystery of pi.

  30. What an excellent article! I’m currently studying trigonometry at school and Radians didn’t make sense to me. However, now I understand them much better.

  31. Dear Kalid,

    I have lots of difficulties understanding solid angles (steradians). For starters, I don’t see how it can be a ratio since it’s an area divided by a length. Furthurmore, I want to know how to calculate solid angles in a 3 dimensional model, but I simply don’t understand many of the explanations provided on the net. I hope you can clear my doubts on this subject.

    Yours faithfully,

    Aadit M Shah

  32. In high-school, when my teacher was attempting to teach the trigonometry required for basic calculus, I had NO IDEA what radians are.

    And then, in a trig book on the internet recently, I saw the explanation: a radian is a what you get when the length of the segment of the circle’s circumference being measured is equal to the length of the radius.


    My immediate reaction was: “Oh. That makes a lot of sense. It simplifies the measuring of angles to the measuring of circumferences.” And it felt a lot more elegant and useful.

    THAT is the feeling of why I love math.

  33. @Dave: Awesome comment — that’s exactly it. One thing I didn’t realize: Suppose I asked you to give me a 37 degree angle. How would you do it? Without tools or a calculator (for fun sine/cosine tricks), you’d have to guesstimate (hrm, I know 45 degrees, maybe I can do a little less).

    But, if I said “Create an angle of 0.5 radians” it’d be straightforward. Take some amount of string, attach it to a pencil, and draw a circle. Then take half of that (0.5) and begin wrapping it around the circumference — whereever it ends is 0.5 radians.

    There are probably much better ways to construct it, but it really helps it click when you realize how hard angles are to draw without external tools.

  34. this is another thank you message,
    this actually helped me understand how exactly mathematics works, i mean the other day i read somewhere about how easier arithmetic would be if we used base 12 rather then 10, and this post came to mind. maybe 100 years from now when computer processing power comes to stagnation we will look for ways to improve our thinking efficiency and say hey base 12, and in no time kids with be struggling to memorize tables in base 12. the same way some mathematics wrote 500 years ago about how easier it would make trig if we used radians but no one cared until we had to calculate angular velocities and stuffs.

  35. @Joy: Heh, a lot of it is based on your reference point for sure. I can’t imagine trying to do math in the days of Roman numerals — once we find a better model (like decimal numbers) the old system seems so antiquated.

  36. Show me a sextant with radian reading and I accept that radians have place also elsewhere than in the nerds’ world. (Or a GPS receiver. Or a compass. Or a milling machine turntable. Or just any practical implementation.)

    I mean, radians are useless in the real life. There are absolutely no practical applications for them, so why bother? Why are the longitudes and latitudes on the map measured in degrees and not in radians? When a sailor or pilot takes a bearing, does he use degrees or radians?

    Nerds may be clever, but they usually fail in practical implementations.

  37. @Ironmistress:
    The article lists some examples of use. For example, if someone says the tire spun 2000 degrees per second, it takes a lot more work mathmatically to find out just how much it’s really spinning, whereas radians are as easy as multiplying it by the radius.

  38. Mike, not any more than making calculations in nautical units than in metric. (The usual units for spinning is revolutions per minute, not radians per second as the SI system would imply).

    That is the reason why the Nautical Almanac lists the various celestial quantities in degrees, minutes and tenths. We are used to them. That is why we use nautical miles instead of kilometres while at sea. One nautical mile is the same as one angular minute of latitude.

  39. @Ironmistress: Great question. I think the main issue is realizing there are several viewpoints we can take.

    When we see an orbiting satellite, do we measure its speed by how fast it goes by our field of view (degrees/sec) or how fast it’s going around its cycle? (radians/sec). When you’re driving around the highway at 60mph, you think that’s your speed… but from the perspective of a gnome at the center of the earth, you are moving .00001 degrees per hour. You care about mph; the gnome cares about degrees per hour.

    In most nautical applications, we’re the “gnome” and want things from our own perspective. That’s fine.

    But the equations of physics/astronomy that enabled the creation of the sextant, etc. are best described with equations in radians (sine, cosine, pendulum motion, etc. are described in radians,

    Utility depends on context — the military uses “mils” (1/1000 of a radian) for firing tables, etc. and not degrees. Hope this helps!

  40. First time I saw you blogs last week. And since last week I kept on reading various articles written by you (adding 1 to 100, birthday paradox, e, natural log etc). I enjoyed all the them. Your ideas and understanding of mathematics is crystal clear!! Wow!!. It helped me to understand mathematics quite natural way.
    Thanks for all your post…
    Please provide some insight into steradian (solid angles). I would like to know how total angle in sphere is 4*pi.

  41. Best math article I have read in years. There is a difference between understanding and teaching and this is incredible stuff. Do you have a place where people can make small donations of gratitude.

  42. @Caleb: Wow, thanks for the kind words! I don’t have a donation page, but I’m more than happy if you help share the site with people who could use it :).

  43. Thank you for this great explanation. I think it was the first time I really believed that radians were more practical than degrees. I radian measure is more useful when dealing with calculus, but now have a better explanation for my algebra 2 and trig. students as to why they should to learn it (other than it will help you in your future math classes).

  44. @Jenn: Glad you enjoyed it! I was the same way, I didn’t appreciate what radians were until understanding their “point of view”, and it turns out, I think it makes more sense than degrees :). Happy to hear it will help with your teaching!

  45. Thanks Kalid for your excellent explanation on Degree and Radian. I was searching for the origin of degree and why do we use radian when I chanced upon your site. Can I have your permission to include this in my school’s Mathematics guidebook? I will acknowledge the source and give credit to you. :)

  46. Hi Kalid,

    You have explained it simple, but i cannot get my head around the Bus example

    “The wheels are turning 6 radians per second”. You’d think:

    Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.

    6 radians per second means , 360 degrees per second , right . But that is 2000 degrees per second.
    Also i dint get the idea of “scale unit cirlce”.
    Sorry to ask , if it is too silly.


  47. Hi, Great way to explain radians! One question regarding the example on how fast the wheel is turning. If we proceed further does we infer that the total no. of rotation of wheel is approx 14 per sec?

  48. @Raju: Whoops, I should have clarified the example. The first gives you degrees per second (2000, a nice round number) and you try to compute it.

    The second gives you radians per second (6, a nice round number) and asks you to compute it. You are correct that 6 radians is close to 360 degrees (I didn’t use the same speed in both examples).

    Radians are measurements on the unit circle (where radius = 1), so you need to enlarge or shrink it depending on your example. In our case, our radius was 2 meters, so we multiplied by 2 (6 radians per second = 6 * 2 = 12 meters per second. If we had a 1-meter radius, then the speed would be 6 * 1 = 6 meters per second).

    @Ajeet: Depending on the example: for 2000 degrees per second, that is about 2000 / 360 = 5.555 rotations per second. For 6 radians per second, that is 6 / (2 * pi) = .95 rotations per second.

  49. this article is so eye opening! My math teacher always recommeds you as a second source after she gives us new material and now i understand why! I’ve never understood what the difference between degrees and radians were untill now! I really that that you put the definitions in simpler terms, “Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled” !I will surely be dropping by again if i ever need other clarifications!

  50. @Melissa: Awesome, so glad it helped! Yep, radians vs. degrees bugged me for years after school, happy you’re able to avoid that headache :).

  51. the Hebrew, Muslim and Mayan calendar have 360 days for number of days in a year. To assume they were in error on their calculations is not wise- consider the accuracy the Egyptians had in building the pyramids- far more accurate than us engineers today. No- there must be another explanation- as possibly in some cosmic event changed the number of days in a year, such as a near miss of the Earth with Venus. It may not be a coincidence that the Hebrew scriptures have a long day back when Joshua fought a battle, and I’m told other ancient calendars noted a long or short day/night at the same time.

  52. hi
    i Should crate a program that this program contain a game.
    this game contain a lable & a picturebox.
    this picturebox shud motion with 30 degrees in the form.
    then picturebox arrived the wall of the form, the picturebox reflexed .
    the angle between first degrees & reflexed degrees shud be 90 degrees.
    i can not solved this program.
    please help me.
    tanx a lot

  53. Dear Kalid

    I really enjoyed reading this article, I really appreciate your effort in making things easy to understand, would love to see more articles from you.


  54. Hi Kalid,
    Thank you for another of your articles that does enogh to rejig my brains, which just runs on autopilot..& thus I have this really nerd kind of question..from the bus example..answer for which I am waiting with great anxiety. You converted 2000 degrees/sec to meters per second in question1, what happened to the conversion of radians per second to meters per second?

  55. @mahir: Great question. Radians don’t have units of their own, and are in terms of the radius (how many “radiuses” you traveled along the outside). Because we measured the radius in meters, the radians are “meters”.

  56. Get yourself elected and in charge of education for your country Kalid. If you’d been my teacher in school I wouldn’t be working in a bar now.

  57. @David: Thanks, though I don’t know if US politics is quite my bag ;). But I’d love to build resources / tools to help teachers.

  58. I guess the most intuitive unit for measuring angles is the turn, cycle or rotation (I don’t know which wold be the best name to pick). One turn (360º), half a turn (180º), 1/4 turn (90º), a 1/5 (72º), a 1/6 (60º) a seventh (7/360 º), 4 turns, 1000 turns, -1/12 of a turn (-30º) and so on. It looks good in decimal format too: 0.8 turns (20% short of a full turn), 0.48 turns (nearly half a turn), -0.987 (almost a backwords turn) , 2.5 turns … Turns per minte is RPMs and per second is Hz. To convet turns to any other angular unit just multiply by the amount of the of new units there are in a single turn: turns to radians mutiply by 2pi, turns to degrees mutiply by 360… To get back to turns devide: 78º is 78/360 turns, 3.5 radians is 3.5/(2pi) turns, pi/2 radians is pi/2/(2pi) = 1/4 turn… It’s a shame that pi is not 2pi so that angles in radians and in turns would be the same apart from the pi constant: 3 turns = 3 (2pi) radians, 1/3 turn = (2pi)/3 radians and son forth. This point is further explained in the Tao manifesto.

  59. @João: Thanks for the comment! I agree, Tau is more natural than pi when thinking about circles :).

  60. Hey Kalid,

    Great article, but I still have a intuitive gap on sin(x)/x

    If we look at the plot

    You explained the first peak, at x=0, but how do you explain the other peaks at x~=8?

    My intuition says that peak represents when we’ve made a full circle and have just barely passed the x-axis again. Is that right?

  61. @Ming: Great question. sin(x)/x is only troublesome around x = 0 because of the division by 0. Otherwise, it looks like a sin(x) wave, which constantly shrinks as we move along [the division by x is shrinking it]. So, all the other peaks are the normal ones, when sine is at its maximum (i.e., 90 degrees, 360+90 degrees, etc.).

    Here’s a graph of sin(x)/x vs regular sin(x):

  62. Hey Kalid
    That was a great article. I didn’t get what radian was at first. Your article helped a lot.
    Wondering if you can help me with this problem. I am a year ten student taking add maths. We are learning differentiation right now. When we are finding the nature of a stationary point, my maths teacher kept saying that if the second derivative is a negative-d2y/dx20 its a minimum point. Why? Is there a logical explanation which involves less proves?

  63. Hi June, thanks — really glad it helped. Great question, though to answer it fully I’d probably need another article, but I’ll try.

    An analogy: The function is your bank account. The derivative is your current weekly income. The second derivative is your weekly raise.

    A stationary point means the function (your bank account) isn’t changing, which means weekly income = 0 [derivative = 0]. That makes sense.

    But what about my weekly raise? If my weekly raise is positive (\displaystyle{\frac{d^2y}{dx^2} > 0}), then my bank account will only increase from now on [next week my income might be $100/week, the week after it might be $200/week, and so on]. Therefore, my current amount is a minimum. There’s more about calculus here:

    Hope this helps!

  64. Degrees and radians are both measurements of central angle, neither one is more selfish or self-centered. How would that dialogue go in radians?

    “Hey Bill, how far did you go?”
    “Well, I had a really good pace, I think I went 6 or 7 mile–”
    “Shuddup. How far did you go in radians?”
    “I’ll use small words for you. Me in middle of track. You ran around. How…much…did..I…turn…my…head?”

    You’d get the same response if you asked how many centimeters he ran. Or inches. :)

  65. Big fan of yours Kalid, please keep these tutorials coming. Your website along khan’s academy are the best resources for understanding math.
    I wish my high-school math teacher explained as you and that I paid more attention.:)

  66. @Mentock: Perhaps, not sure though. Far easier to count distance from your perspective [even number of paces or steps taken] than to figure out how far the observer moved their head.

  67. I was just about to congratulate you on the clarity of your explanation and to finally help me make sense of the difference between degrees and radians, but before I do, I guess I should make sure that I got this right: if I understand correctly, one way to put it would be that a degree is in fact an angular measure where a radian is more like a distance unit? Also, 45 degrees will always be 1/8 of a complete circle and nothing wil ever change that, but the radian value will vary based on the radius of that circle, right?

    If this is correct, then kudos for your article!

  68. Hi Phil, great comment. 99% of the way there.

    Radians are divided by the radius of the circle to “normalize” them. So, they technically have “no unit” (they’re distance traveled / size of radius or distance / distance). But in our heads, we could think of them as “distance on a unit circle”.

    Alice: “I traveled 3 miles around my track. I got a different view”.
    Bob: “How far would that be on a unit circle?”
    Alice: “Oh, that’d be quarter of the way around”.

    It’s easier to describe things relative to the unit circle (a reference), not the circle she happened to have lying around (pi/2 is quarter of the way around on a unit circle).

    It’s a bit like comparing strengths between different sized animals. Is a bear stronger than a lion? No idea. If they both weighed 100lbs (a “reference weight”), now which is stronger?

    Radians are “How far did you go, if you were on a unit circle?”. A raw distance is hard to compare.

    Hope that helps!

  69. Your writing is hilarious, simple, and so easy to understand. Other teachers (especially math) should learn your style. I love your examples, too. Thank you and keep up the excellent work.


  70. hey… ur explanation is really good. i have a presentation on radian angle tomorrow… do u knoe where i can find some more funda on radian

  71. Kalid, I’m just fixing to get started back in school. After 10 years of being out of high school and working in mechanics on heavy equipment use numbers and math formulas on a very regular basis. Just took a pre-placement test for the school I’m going to to enrolled in…. WOW have I lost my edge in math. I did some looking on google and stumbled across several forums and sites, including yours. Have to say your explanation made sense to me faster than any other. Thanks for your time of writing and responding on here. I’m sure I’ll be using your input more and more in the near future.

  72. Hey Justin, thanks for for the note. Really glad the material is clicking for you, my goal is to get us (including me!) from 0 to “aha!” in the shortest time possible. Really gratifying to hear it’s working.

  73. I dropped out from architecture when I was a teenager because the Calculus lectures were so utterly cryptic -and no teacher I knew would make it any clearer- that this was a source if horrible stress I simply could not cope with.

    Now, at 40, I am considering taking back my original wish of being an architect, and by having seen your explanations -addresses for right brain people like myself, I now have hope.


  74. Indeed, you have a gift here, Kalid
    I feel exactly as everyone else who’s praising you here

    I’ve understood complex numbers in 1 day,
    a feat no school could help me achieve 😀

    specific to this article,
    I believe it would bring even more clarity and click!ness
    if the link between radius and radian would be emphasized
    this created the biggest impact in me when I read about it
    exactly what Dave said in a comment in 2010

    Thank you

  75. Thanks Michael! Really glad it helped. Great feedback on the link between radian/radius, I’ve updated the article. In short, I see “radians” as shorthand for “distance traveled in terms of radius units”.

  76. Hi.
    I’m still having a hard time understanding how to work with radians. Or what they are exactly.
    But you definitely have a great way of teaching.
    No doubt.
    I guess I’m just not cut out for maths.

  77. I was under the impression that when six equilateral triangles are brought together such that they hav a common vertex, the 6th triangle’s leading edge coincides with the 1st triangle’s lagging edge, n since the Mesopotamians considered the number 60 as their base (sexagesimal no. system instead of decimal), so the total measure of the angles at the common vertex would be 6*60=360.

    Now from ur article i hav to guess that the Mesopotamians seemingly went the other way round, as in, they divided the circle into 360 parts based on the approximate annual cycles, n found that six equilateral triangles form the said 360 divisions, then calculated the no. of divisions in the angles of the triangle to be 360/6=60, n hence derived the sexagesimal number system..

  78. Why does everything have to be so complicated and complex? I’m glad God invented calculators!

  79. Dear Mr. Khalid,

    I didn’t understand the problem of bus wheel 2000degree/sec. How 2000deg/sec is converted to 6 rad/sec? please explain that. if possible in my mail.ID

    Thanks in advance

  80. Your Example 1 Wheels on the Bus is incorrect. Rim size is the inside diameter of the tire, not the outside diameter of the tire.

  81. The wheels are turning 2000 degrees per second”. You’d think:

    Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second. Circumference = 2 * pi * r, so it’s moving, um, 2 * 3.14 * 5 and 5/9ths… where’s my calculator…
    “The wheels are turning 6 radians per second”.
    Sir, give the calculation to get 6radians .

  82. Hi Srinivasan, the two numbers [2000 degrees and 6 radians] weren’t meant to be equivalent, I was picking nice, round numbers in each case.

    However, you can convert between them using the relationship:

    360 degrees = 2 * pi radians, or

    1 radian = 360 / (2 * pi) = 57.3 degrees

  83. Thank you for this wonderful explanation. I’m not sure about one of your calculation about the truck speed:

    radius of wheel: 2 meters;
    speed: 2000 degrees / second


    You said: Circumference = 2 * pi * r, so it’s moving, um, 2 * 3.14 * 5 and 5/9ths… where’s my calculator…
    It is not quite right because in this case ‘r’ mean ‘radius’ not ‘radians’. Then:

    circumference of wheel(meters): 2 * PI(3.1416) * 2 meters = 12.5664 meters
    circumference of wheel(radian): 2 * PI

    You said: “The wheels are turning 2000 degrees per second”. You’d think:

    Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second.

    “The wheels are turning 6 radians per second”. You’d think:
    speed(rotations / second): 2000 / 360 = 5.5555

    You said:
    Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.
    It is not the right answer since you calculate Rotation * Radius and it should be Rotation * Circumference.

    speed(meters / second): rotation(5.5555) * circumference of wheel(12.5664) = 69.8126 meters / second
    speed(radians / second): 2 * PI * rotation(5.5555) = 34.90623 rad/sec

    speed(km / h) = (meters / second(69.8126) * 60(seconds) * 60(minutes)) / 1000(meters) = 251.32 KM/h

    Then the wheel speed is not 12m/sec as you said but 70m/sec(34.90623 rad/sec).

    OR maybe I miss something along the way.

    Great article! I quickly and finally understood Radians. I can now use it knowing what I do. Thank you very much! Nice work.

  84. Am in my first at a university in Zambia & am trying to understand what radians are because i can’t understand pi and it’s relation to measurement…what’s pi??ofcourse don’t say 3.142…

  85. Hi
    First thanks for the article it is a really great article like all the articles.
    However I do not fully understand why you consider degree as angle measurement but radian you consider as distance measurement when in fact both are the same as even degree was meant as specifiyng the length of the arc, and it has then been adopted as a method of specifying the steepness of the line that connects the end of the arc segment with the center, and this has been generalized to any two lines and the angle between them, and this holds true with radians as well.
    on the other hand distance can be measured just as well with degrees by multiplying it by the diameter, something that was the true intention to begin with.
    In fact degrees and radians are just the same thing, and this is arc length of a unit circle, they are just different systems of measurement just as US and the metric system differ or the fraction vs decimal point systems.
    But in reality the truth is that degrees lend themselves more natural toward angle measurement with the more precise and clear portions, while radian measurement is more natural when dealing with total distance or total area, which are the primary concerns in mathematics and physics.
    And we say more natural we mean just as the metric system is more natural for calculations, and the fractional system is more natural for everyday activities in which we use querters and thirds, while the decimal point system is more natural for currency and financials.

  86. @Mike I am not sure what calenders you speak about, and what is the point that you are trying to make, but well the Muslim and other calendars are not based on the sun at all, leaving only the Christian and Jewish calendars being in sync with the sun, and none of them has 360 days.
    The Christian calendar is 365-366 based, and was definitely wrong two thousand years ago and as a matter of fact have been updated from the Julian to the gregorioan calendar.
    The Jewish calendar is more complicated is the months are in sync with the moon (since the actual source of the month is the rebirth of the moon), but the year is kept in sync with the sun (since the concept of a year is based on the sun), which is achieved by having a simple year of around 353-355 days and a leap year of 364-366 days, setup in cycles of 19 years, which incident turns out to be fully in sync even with modern astronomy without any changes over the course of thousands of years!!

  87. I was blind how 2π radians equals to 360° but this helps me see what I couldn’t before it helps a whole lot

  88. @comment #55: Sorry Ironmistress, but (nautical) mile is same as one angular minute of latitude just around the 45th degree latitude (I live very near to that latitude and at low altitude, but I am not so selfish to think only as I am pleased … :-) ).
    @comment #56: “the military uses “mils” (1/1000 of a radian) for firing tables …”. In the former Jugoslavija (“Suomi” means “Finland” and “Deutchland” means “Germany”, OK :-) ) we used two kinds of “the thousandth”: our compasses were calibrated as 360 degrees == 6400 “thousandths”, but Warsaw pact made anti-aircraft guns on truck (nicknamed “Praha” after the capital of (then existid) Chehoslovakia) which had 6000 “thousandths” so these generations had to recalculate “NATO thousandths” into “Warsaw pact thousandths” (and vice versa, as well … :-) ). When my officer asked a question “How many thousands are there in a whole circle?” I volunteered my “wisdom” by saying “Mr. , there are approximately 6283 thousandths, but, because of practical reasons, NATO uses 6400 and Warsaw pact 6000 thousandths.” I was such a kid …
    @comment #86: Sorry, Khalid, but extrema of sin(x)/x and sin(x) are not exactly the same (but are near). The first derivative of sin(x)/x is (x*cos(x)-sin(x))/x^2, so extrema may be calculated as zeroes of tan(x)==x, which is a little smaller than (k+1/2)*pi, “k” being a positive integer (there are also symetrical points (negative) …). Here are positions of the first thirteen ( :-) ) extrema of sin(x)/x (recalculated into degrees in order to see how far are they from positions of extrema of sin(x), approximately rounded, of cause): 257.453397562 (not 270), 442.624325932 (not 450), 624.760150382 (not 630), 805.933545018 (not 810), 986.676597109 (not 990), 1167.18968275 (not 1170), 1347.56536463 (not 1350), 1527.85236530 (not 1530), 1708.07879126 (not 1710), 1888.26199997 (not 1890), 2068.41329240 (not 2070), 2248.54034368 (not 2250) and 2428.64854982 (not 2430). Sory for being so detailed …
    Beyond comment #86 those “kids” may really learn nice things on your site …

  89. I like the universal radian measure concept. like from the sun’s perspective He (the sun) knows: As my light travels to the Earths surface i fully encompass the surface of any spherical (like me) shaped objects along the way. So theres further away spherical objects needed (and of course in place) to make sure Earth gets its whole surface sun light bathed.
    BUT….dont forget.
    6 radians is 6 radians; however, THINK: hanging Fire Sprinklers… you need to measure what would the measure of the radian be that woulf ensure this sprinkler would cover the right amount of surface area. And you measure that in inches. or any linear unit.
    I do realize here you have the “Intuitive” guide. but there is a practical application as well in Fire Sprinkler Fitting.

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