Degrees and radians is one of 1 angles formula in the trigonometry section of this library, and it is used at high school level.
Why degrees and radians works
One full turn is 360 degrees and also 2π radians, so 180 degrees and π radians name the same half turn. That makes the fraction π/180 equal to 1, and multiplying by 1 never changes a quantity, only its label. Flipping the fraction converts the other direction.
What each symbol means
The subscripts identify the angle unit.
Degrees and radians: when it holds
Keep $\pi$ for an exact radian answer.
When it stops applying
Converting the answer afterwards does not repair a calculation that needed radians from the start. Derivative and series rules such as the derivative of sin x being cos x are only true in radians, so a degree-mode derivative is off by the factor π/180, about 0.01745, and no later conversion fixes it.
Degrees and radians: a worked example
$150^\circ=150(\pi/180)=5\pi/6$.
The mistake to avoid
What people do: Students convert the number correctly but leave the calculator in the other mode.
Why it goes wrong: The calculator reads whatever you type through its current setting, so a value meant as radians gets treated as degrees or the reverse, and the output looks plausible while being badly wrong.
Do this instead: Test the mode before you start: sin 30 should return 0.5. If it returns −0.988, the calculator read your 30 as radians, which is nearly five full turns around the circle.
Degrees and radians: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. The subscripts identify the angle unit.
- Check the conditions before substituting. Keep $\pi$ for an exact radian answer.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Angles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where degrees and radians comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Unit Circle — the lesson behind this formula: connect angles to coordinates, sine, and cosine.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about degrees and radians
Should I multiply or divide by π/180?
Multiply when you are going from degrees to radians, since radians are smaller numbers for the same angle. Sanity check with 180 degrees: it must come out as π, about 3.14, not 10313.
Why do mathematicians prefer radians?
Because a radian measures arc length on a circle of radius 1, so the angle and the distance travelled along the arc are the same number. Degrees are an arbitrary split of a circle into 360 parts inherited from ancient astronomy.
How do I convert 5π/6 back into degrees?
Multiply by 180/π and the π symbols cancel: 5(180)/6 = 150 degrees. Whenever a π sits in the angle, expect the cancellation, and you rarely need a decimal at all.
Is it fine to leave π in my answer?
Yes, and usually it is preferred. Writing 5π/6 is exact, while 2.618 is rounded, and exact forms keep later work clean when more π terms show up and cancel.