# Difference Between Degrees on a circle

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

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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

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 $360+a-b$.

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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:
Trigonometry
Euclidean geometry
Circle

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@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.

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