Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What kind of math would I use to calculate the difference between two degrees on a circle? Say, 38 and 272 degrees? When I just subtract one position from another sometimes it's more than 180 or sometimes I cross over 0/360. I need keywords that can help me learn more about it. Ultimately I want to create an excel formula but I don't know what it is called. Thanks

share|improve this question
Maybe you're looking for the modulo operation? –  J. M. Aug 23 '12 at 11:54
yes may be modulo $2\pi$? –  Seyhmus Güngören Aug 23 '12 at 11:58
@Seyhmus or $\bmod 360$, if you want to keep things entirely in degrees. –  J. M. Aug 23 '12 at 12:03
@J.M. sure it seems more interesting) –  Seyhmus Güngören Aug 23 '12 at 12:10

3 Answers 3

If $a$ and $b$ are angles in a circle, measured in degrees, with $0\le a\le b\lt360$, then the difference between them is the smaller of $b-a$ and $360+a-b$.

share|improve this answer

Points on a circle whom angles differ only by a $2k\pi$ (where $k$ can be any positive or negative integer) are the same. So $25$ and $25+360=385$ and $25-360=-335$ are in the same position on the circle. It is said that $25$, $385$ and $-335$ are congruent modulo $360$ e.g. $$ 25 \equiv 385 \mod 360 $$ There is a convention which considers counter-clockwise rotation to add a positive angle to the starting angle and clockwise ones to add a negative angle to the starting angle.
Note: $\pi$ radians is equal to $180$ degrees
These are usually discussed in Elementary Geometry or Elementary Trigonometry
See these pages for further details:
Euclidean geometry

share|improve this answer
@GerryMyerson Oh, fixed it. ;) –  Zeta.Investigator Aug 23 '12 at 12:23
@GerryMyerson: Is it mathematically correct of saying that $\pi$ is equal to $180$ in degrees? –  Babak S. Aug 23 '12 at 12:52

The following aspect has not be addressed in the answers so far: One has to distinguish between oriented and nonoriented angles.

The nonoriented angle $\phi$ between two points ${\bf u}$, ${\bf v}\in S^1$ (the unit circle) is the length of the shorter arc on $S^1$ connecting ${\bf u}$ and ${\bf v}$. It is a number between $0$ and $\pi$ (inclusive) and is given by the formula $\phi=\arccos({\bf u}\cdot{\bf v})$, where the $\cdot$ denotes the scalar product in ${\mathbb R}^2$. In terms of everyday geometry it is the angle between the "rays" ${\bf u}$ and ${\bf v}$ as measured in degrees by a protractor.

The oriented angle is a notion connected to rotations and can be any real number; but depending on the problem at hand it may be restricted, e.g., to the interval $\ ]{-\pi},\, \pi[\ $. Let ${\bf u}_0=(\cos\alpha,\sin\alpha)\in S^1$ and a $\phi\geq0$ be given. Then the motion $$t\mapsto {\bf u}(t):=\bigl(\cos(\alpha+t),\sin(\alpha+t)\bigl)\qquad(0\leq t\leq\phi)$$ turns ${\bf u}_0$ continuously counterclockwise to a final position ${\bf u}(\phi)$. The total turning angle is $\phi$. A similar definition for $\phi\leq0$ produces a clockwise rotation. The final position ${\bf u}(\phi)$ coincides with the starting position ${\bf u}_0$ iff $\phi$ is an integer multiple of $2\pi$.

When dealing with geometrical (or kinematic) problems where angles are at stake one always should assert whether the introduced variables denote nonoriented or oriented angles.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.