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”.

This is a companion discussion topic for the original entry at

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:

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).

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)?

@Fredg: Great catch! Yes, that was my mistake. Fixing it up now.

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. :slight_smile:

@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.

You are the only one I know who can make a math blog post sound funny!

Thanks Siya – I think there are gems hidden away in almost any topic :).

[…] According to Better Explained, degrees are subjective but radians are objective. 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? […]

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

Thanks for the encouragement Sid! I hope to increase the output too :).

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.


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

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


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.


@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.

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!!!

(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,

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 :slight_smile: