Angle Converter
Enter an angle in degrees or radians. The converter gives the equivalent in the other unit, the coterminal angle in the range 0°–360°, the quadrant it lands in, and the reference angle used to relate it to first-quadrant trig values.
What this converter reports
Degrees and radians measure the same thing in different units. A degree is one three-hundred-and-sixtieth of a turn, inherited from Babylonian astronomy; a radian is the angle subtending an arc equal to the radius, which is why the calculus of trigonometric functions is clean in radians.
This converter moves an angle between the units and reports three quantities that depend only on where it points: the coterminal angle in [0°, 360°), the quadrant, and the reference angle.
The conversion is the easy half. The useful half is the reduction: any angle, however large or negative, points the same way as exactly one angle in [0°, 360°), and that representative fixes the sign and magnitude of every trigonometric value.
The reference angle then reduces the problem one step further. Every trigonometric value of any angle equals the value at its reference angle, up to a sign fixed by the quadrant. That is why a table of first-quadrant values is enough to evaluate the whole circle by hand.
How to use this calculator
- Enter the angle value Any real number is accepted, including negatives and values well beyond a full turn. There is no need to reduce the angle yourself first — reducing it is one of the things the tool does.
- Select the unit of what you typed The dropdown declares the unit of your input, not the unit you want out. Both units are always reported; choosing “Degrees” means the number you entered is in degrees.
- Read the conversion line first The first step shows the multiplication actually performed: π/180 going from degrees to radians, 180/π coming back. Radian results are decimals, not multiples of π.
- Then read the reduction lines The coterminal angle, quadrant and reference angle follow, all computed from the degree value regardless of which unit you entered.
The formula, and where it comes from
rad = deg × π/180 deg = rad × 180/π reference = distance from the nearest horizontal axis
The conversion factor follows from a single identity: a full turn is 360 degrees and also 2π radians, so 180 degrees equals π radians. Every conversion is that one proportion applied.
The coterminal angle is the degree measure modulo 360, with 360 added to a negative remainder so the result lands in [0, 360). The reference angle is then measured to the horizontal axis — θ in quadrant 1, 180 − θ in quadrant 2, θ − 180 in quadrant 3, 360 − θ in quadrant 4 — and is always between 0° and 90°.
What each input means
- θ Angle value — form field “Angle value”
- The angle to convert. Positive values are measured anticlockwise from the positive x-axis, negative values clockwise, following the standard mathematical convention rather than the clockwise bearings used in navigation. Units: degrees or radians, as chosen in the unit selector.
- unit Input unit — form field “Input unit”
- Declares how to interpret the number you entered. This is the single most common source of a wrong answer here: entering 3.14159 while the selector still says degrees converts a tiny angle, not half a turn.
- θ mod 360 Coterminal angle
- The unique angle in [0°, 360°) pointing the same way. Angles differing by a whole number of full turns share every trigonometric value, so this is the representative used for the quadrant and reference angle. Units: degrees.
- θ′ Reference angle
- The acute angle between the terminal side and the horizontal axis, always in [0°, 90°]. It carries the magnitude of every trigonometric value; the quadrant supplies the sign. Units: degrees.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
A third-quadrant angle
Convert 210° and locate it. This is a standard exam angle whose sine and cosine are expected from memory, so it is a good test of what the reference angle buys you.
Inputs Angle value = 210, Input unit = deg
- To radians 210° × π/180 = 3.66519 rad
- Coterminal angle 210° ≡ 210° (mod 360°)
- Quadrant Quadrant 3
- Reference angle 30°
Result 210° = 3.66519 rad · reference angle 30°
The angle is already inside one turn, so the coterminal line simply confirms it. The reference angle of 30° is the payoff: values at 210° match those at 30° in magnitude, and the third quadrant makes sine and cosine negative, giving sin 210° = −1/2 with no evaluation.
The radian figure appears as a decimal, not 7π/6 — the conversion is numeric and does not recognise rational multiples of π.
A negative angle, and what the coterminal line is for
Convert −45°. A negative angle is measured clockwise, which puts it below the positive x-axis, and its coterminal representative is what makes the quadrant unambiguous.
Inputs Angle value = -45, Input unit = deg
- To radians -45° × π/180 = -0.785398 rad
- Coterminal angle -45° ≡ 315° (mod 360°)
- Quadrant Quadrant 4
- Reference angle 45°
Result -45° = -0.785398 rad · reference angle 45°
The reduction adds a full turn to reach 315°, putting the angle in the fourth quadrant with a reference angle of 45°. Cosine is then positive and sine negative, as expected for a clockwise 45°.
The radian line keeps the negative sign because it converts the angle as entered, not the reduced form. Both lines describe one direction: one preserves how you wrote it, the other normalises it.
Reading the result
Radian output is a decimal
Results are numeric to six significant figures, so π appears as 3.14159 rather than as a symbol. If you need an exact multiple of π, divide the reported radian value by 3.14159 and look for a simple fraction — 3.66519 divided by π is 7/6, for instance.
Angles on an axis
When the reduced angle is a multiple of 90° the tool says the angle lies on an axis instead of naming a quadrant. That is not evasion: 90°, 180° and 270° are boundaries belonging to no quadrant, and at those angles one of sine or cosine is zero while the other is ±1.
Rounding near a boundary
A rounded radian input leaves a small residue: 3.14159 rad reports a reference angle near 0.00015° rather than zero, because 3.14159 is not quite π. That is your input's precision showing, not a conversion error.
When you would use this
Feeding a programming language's trig functions
Almost every standard library — JavaScript's Math.sin, Python's math module, C's math.h — expects radians. Converting first fixes the classic bug where sin(30) returns −0.988 instead of 0.5.
Evaluating trigonometric values by hand
Reducing 1290° to its coterminal 210° and reference 30° turns an intimidating evaluation into a table lookup plus a sign — the standard exam technique, performed here exactly as you would on paper.
Assumptions and limitations
What this calculator assumes
- Angles are measured from the positive x-axis, anticlockwise for positive values — the standard-position convention of mathematics.
- The unit selector correctly describes the number entered; the tool cannot infer the intended unit from the magnitude.
Where it stops being the right tool
- No gradians, turns, or degrees-minutes-seconds. Only decimal degrees and radians are supported, so 30°15′ must be entered as 30.25.
- Radian output is never symbolic: exact forms such as 7π/6 are not recognised or produced.
- No trigonometric values are evaluated. The tool locates the angle and gives its reference angle; computing sine or cosine is a separate solver.
Common mistakes
Leaving the unit selector on the wrong setting
Why it happens. The default is degrees, so pasting a radian value straight in converts it as if it were degrees. The result looks plausible — a small number of degrees is a perfectly ordinary angle — so nothing signals the error.
How to avoid it. Check the selector before reading the answer. A quick sanity test: any angle you meant in radians and entered as degrees will come back as a suspiciously tiny fraction of a radian.
Confusing the reference angle with the coterminal angle
Why it happens. Both are reductions and both are reported together, so they are easy to interchange. The coterminal angle can be anything up to 360°, while the reference angle never exceeds 90°.
How to avoid it. Use the range as the test. If the value you are about to look up in a trigonometric table exceeds 90°, it is the coterminal angle and you have taken the wrong line.
Measuring the reference angle to the vertical axis
Why it happens. In the second and third quadrants the terminal side can look closer to the vertical, which tempts a subtraction from 90° or 270° instead of from 180°.
How to avoid it. The reference angle is always measured to the horizontal axis. For 210°, subtract 180 to get 30 — not 210 − 90.
Key terms
Frequently asked questions
How do degrees and radians relate?
A full turn is 360 degrees and also 2π radians, so 180° = π radians and 1° = π/180 radians. Every conversion here is that single proportion applied in one direction or the other.
What is a reference angle, and why does it matter?
It is the acute angle between the terminal side and the horizontal axis, always between 0° and 90°. Every trigonometric value of an angle has the same magnitude as at its reference angle, with the sign supplied by the quadrant — which is why a first-quadrant table covers the whole circle.
What is a coterminal angle?
An angle pointing in the same direction, differing by a whole number of full turns. The tool reports the unique coterminal angle in the range 0° to 360°, which is what determines the quadrant.
Why is the radian answer a decimal rather than a multiple of π?
The conversion is numeric and reports six significant figures, so 210° comes back as 3.66519 rather than 7π/6. Dividing the result by π will reveal the exact fraction when there is one.