Angle Conversion Calculator Guide: Degrees, Radians, Gradians, and Turns
Angles can be expressed in several measurement systems. Degrees are common in geometry and everyday applications, radians are standard in higher mathematics and calculus, gradians divide a right angle into 100 units, and turns describe an angle as a fraction or multiple of a complete revolution.
All four units describe the same geometric quantity. The conversion depends on the fact that one full revolution equals 360 degrees, 2π radians, 400 gradians, or 1 turn.
This means a half-turn equals 180 degrees, π radians, or 200 gradians. A quarter-turn equals 90 degrees, π/2 radians, or 100 gradians.
Radians are especially important because they connect angle directly to arc length. An angle of one radian subtends an arc whose length equals the radius of the circle.
That geometric definition is why radians appear naturally in calculus, trigonometric derivatives, oscillations, rotational motion, and many scientific formulas.
Degrees remain intuitive because a complete circle is divided into 360 equal parts. They are common in navigation, surveying, geometry, construction, and everyday descriptions of direction or rotation.
Gradians divide a full turn into 400 units, so a right angle equals exactly 100 gradians. This decimal structure has historically made gradians useful in some surveying and engineering contexts.
Turns are perhaps the simplest conceptual rotation unit: one turn is one complete revolution, 0.5 turn is half a revolution, and 2 turns represents two complete rotations.
The calculator should preserve exact radian forms involving π whenever practical. For example, 60 degrees is exactly π/3 radians. Returning only 1.047197551... loses the exact mathematical relationship.
This page converts the measurement unit only. It does not evaluate sine, cosine, tangent, or other trigonometric functions. Those calculations belong in the Scientific Calculator.
How to Convert Angles Between Degrees, Radians, Gradians, and Turns
- Enter the angle: Use a positive, negative, fractional, or decimal angle value.
- Choose the input unit: Select degrees, radians, gradians, or turns.
- Choose the output unit: Select the angle unit you want to convert into.
- Preserve exact π notation when appropriate: For common degree-to-radian conversions, show exact values such as π/6, π/4, π/3, π/2, and π when possible.
- Show a decimal approximation: Provide a numerical approximation alongside exact π notation when it improves usability.
- Keep angle direction: Negative inputs remain negative after conversion because the unit changes but the rotational direction does not.
- Allow angles beyond one revolution: Values greater than 360 degrees or 2π radians remain valid and should not automatically be reduced unless normalized-angle mode is explicitly selected.
Formula and variables
Every angle-unit conversion is based on equivalent representations of one full revolution. Conversion factors are ratios equal to one, so multiplying by the appropriate ratio changes the unit without changing the angle itself.
360° = 2π rad = 400 grad = 1 turn; Degrees → Radians: rad = degrees × π/180; Radians → Degrees: degrees = radians × 180/π- D — Degrees
- Angle measured so one full revolution equals 360 degrees.
- R — Radians
- Angle measured so one full revolution equals 2π radians.
- G — Gradians
- Angle measured so one full revolution equals 400 gradians.
- T — Turns
- Angle measured so one complete revolution equals one turn.
- π — Pi
- The mathematical constant relating a circle’s circumference to its diameter.
Scenario 1: Convert 120 Degrees to Radians
A user wants to express 120 degrees in radians.
- Angle
- 120
- From
- Degrees
- To
- Radians
- Use radians = degrees × π/180.
- 120 × π/180.
- Reduce 120/180 to 2/3.
- Exact result = 2π/3 radians.
- Decimal approximation ≈ 2.094395102 radians.
Result: 120° = 2π/3 rad ≈ 2.094395102 rad.
Both forms represent exactly the same angle. The π-based result is exact; the decimal form is an approximation.
Understanding your results
Converted angle
This is the same geometric angle expressed in another unit.
Only the numerical representation and unit change.
Exact radian form
When the angle corresponds to a rational multiple of π, the calculator can preserve that exact form.
This is especially useful in trigonometry and symbolic mathematics.
Decimal approximation
This gives a finite numerical representation of the converted angle.
Values involving π are generally irrational and therefore require approximation in decimal form.
Turns
Turns show how much of a complete revolution the angle represents.
For example, 90° = 0.25 turn and 540° = 1.5 turns.
Assumptions
- Angles are interpreted as directed real-valued rotations.
- One complete revolution equals 360 degrees, 2π radians, 400 gradians, and one turn.
- Negative angles preserve their sign during conversion.
- Angles are not automatically normalized to one revolution unless a normalization option is selected.
- Exact π forms are simplified using rational reduction where practical.
- Decimal results are approximations when the exact result contains irrational constants.
Limitations
- Angle conversion changes units but does not calculate trigonometric function values.
- A decimal radian value involving π is usually an approximation rather than an exact representation.
- Normalizing an angle and converting an angle are different operations.
- Angles separated by whole turns are coterminal but are not numerically identical before normalization.
- Direction matters for signed angles: positive and negative rotations are distinct even when they can terminate on the same ray after whole revolutions.
- Degree-minute-second notation requires an additional parsing and conversion layer beyond ordinary decimal degrees.
- Bearings and azimuths can use conventions different from ordinary mathematical angles and should not be treated as simple degree values without accounting for direction notation.
- Angular velocity units such as radians per second are rates and require a separate unit conversion from static angle measures.
Common mistakes
- Multiplying degrees by 180/π instead of π/180.
- Multiplying radians by π/180 instead of 180/π.
- Using 360 radians for one full revolution.
- Assuming π radians equals 360 degrees instead of 180 degrees.
- Treating radians as dimensionless numbers without recognizing the active angle context in a calculator.
- Changing the sign of a negative angle during conversion.
- Reducing every angle automatically to between 0 and 360 degrees.
- Confusing gradians with degrees.
- Confusing a turn with 100 degrees rather than one full revolution.
- Rounding π to 3.14 before the final calculation.
Practical use cases
Scenario 2: Degrees to radians
Convert 45°.
45 × π/180 = π/4 rad.
Scenario 3: Radians to degrees
Convert π/6 radians.
(π/6) × 180/π = 30°.
Scenario 4: Degrees to turns
Convert 270°.
270 ÷ 360 = 0.75 turn.
Scenario 5: Turns to radians
Convert 0.25 turn.
0.25 × 2π = π/2 radians.
Scenario 6: Degrees to gradians
Convert 90°.
90 × 400/360 = 100 grad.
Planning and decision guide
One full revolution defines all unit relationships
360° = 2π radians = 400 gradians = 1 turn.
Every conversion can be derived from this identity.
Scenario 7: Half revolution
Half of 360° is 180°.
Half of 2π is π radians.
Half of 400 is 200 gradians.
Half of one turn is 0.5 turn.
Degree-to-radian conversion uses π/180
Because 180° and π radians represent the same angle, multiplying degrees by π radians per 180 degrees cancels the degree unit.
The remaining unit is radians.
Scenario 8: Convert 30°
30 × π/180.
Reduce 30/180 to 1/6.
Result = π/6 rad.
Radian-to-degree conversion reverses the factor
Multiply radians by 180/π.
The radian relationship cancels and leaves degrees.
Scenario 9: Convert 2π/3 radians
(2π/3) × 180/π.
Cancel π and simplify.
Result = 120°.
Radians arise naturally from arc length
For a circle of radius r and arc length s, the angle in radians is θ = s/r.
An angle of one radian therefore subtends an arc equal in length to the radius.
Scenario 10: Arc equals radius
If s = r, then θ = s/r = 1.
The angle is exactly 1 radian.
A full circle has circumference 2πr
Using θ = s/r with s = 2πr gives θ = 2π.
This is the geometric reason a full revolution equals 2π radians.
Radians simplify calculus formulas
Standard derivative identities such as d/dx sin(x) = cos(x) assume x is measured in radians.
Using degrees introduces additional conversion constants.
The Scientific Calculator should make DEG/RAD state visible
Angle Conversion changes the unit explicitly.
The Scientific Calculator uses the selected unit when evaluating trigonometric functions.
Scenario 11: Same angle, different trig input
sin(30°) and sin(π/6 rad) are the same geometric calculation.
Both equal 0.5 when the angle mode is interpreted correctly.
Degrees divide the circle into 360 parts
A right angle is 90°, half-turn is 180°, and full turn is 360°.
Degrees are especially common in elementary geometry and applied measurement.
Gradians divide a right angle into 100 parts
Because a right angle is 100 grad, a full turn is 400 grad.
This produces convenient decimal subdivisions in some applications.
Scenario 12: 45° in gradians
45° is half of a right angle.
Therefore it equals 50 grad.
Degree-to-gradian conversion uses 10/9
400/360 simplifies to 10/9.
Therefore grad = degrees × 10/9.
Gradian-to-degree conversion uses 9/10
Degrees = gradians × 9/10.
For example, 200 grad = 180°.
Radians to gradians use 200/π
Since π radians = 200 gradians, multiply radians by 200/π.
The reverse conversion multiplies gradians by π/200.
Turns make revolution fractions explicit
A turn is dimensionally intuitive for rotation.
One quarter turn, half turn, and three-quarter turn correspond to standard geometric orientations.
Scenario 13: 3/4 turn
3/4 × 360 = 270°.
3/4 × 2π = 3π/2 radians.
3/4 × 400 = 300 grad.
Turns to degrees multiply by 360
degrees = turns × 360.
Degrees to turns divide by 360.
Turns to radians multiply by 2π
radians = turns × 2π.
Radians to turns divide by 2π.
Turns to gradians multiply by 400
gradians = turns × 400.
Gradians to turns divide by 400.
Common angles should preserve exact π forms
30°, 45°, 60°, 90°, 120°, 180°, and many other rational-degree angles have simple exact π-based radian forms.
Exact representation is usually more useful mathematically than a long decimal.
Scenario 14: 225°
225 × π/180.
Reduce 225/180 to 5/4.
Result = 5π/4 radians.
Fraction reduction creates the clean exact radian result
The coefficient degrees/180 can be simplified before attaching π.
The Fraction Calculator provides the underlying rational simplification logic.
Scenario 15: 150°
150/180 simplifies to 5/6.
Therefore 150° = 5π/6 rad.
Not every decimal-degree input yields a neat π fraction
A value such as 17.3° corresponds exactly to 173π/1800 radians if the decimal input is interpreted exactly.
The calculator can preserve that rational multiple of π if desired.
Decimal radians often hide exact structure
1.57079632679 is approximately π/2.
If the original source was π/2, preserving π/2 is more informative.
Do not guess exact π forms from arbitrary approximations too aggressively
A decimal close to π/3 might be a rounded measurement rather than an intended exact mathematical constant.
Exact-form recognition should use a documented tolerance and preferably be optional.
Negative angles represent opposite rotational direction
In standard mathematical convention, positive angles are usually counterclockwise and negative angles clockwise.
Conversion preserves that direction by preserving the sign.
Scenario 16: Convert −90°
−90 × π/180 = −π/2 radians.
The angle remains negative.
Angles larger than one turn are valid
An angle can represent multiple revolutions.
For example, 720° equals 4π radians or 2 turns.
Scenario 17: 450°
450° = 5π/2 radians.
It is also 1.25 turns.
Coterminal angles differ by complete turns
Angles differing by 360°, 2π radians, 400 grad, or one turn have the same terminal direction.
They are still different numerical angle measures unless normalized.
Scenario 18: 30° and 390°
390° − 30° = 360°.
They are coterminal but not numerically identical.
Normalization is separate from conversion
Conversion changes units.
Normalization chooses an equivalent representative within a selected interval.
A normalized degree angle commonly uses 0° ≤ θ < 360°
A common formula is ((θ mod 360) + 360) mod 360.
The extra adjustment handles negative inputs under programming-language modulo conventions.
Scenario 19: Normalize −30°
Add one full revolution.
The equivalent normalized angle is 330°.
Signed principal-angle normalization can use another interval
Applications sometimes prefer −180° < θ ≤ 180° or −π < θ ≤ π.
The chosen interval should be explicit.
Degree-minute-second notation is another representation
One degree contains 60 arcminutes and one arcminute contains 60 arcseconds.
This is a subdivision of degrees rather than a new full-circle unit system.
Scenario 20: 30° 30′
30 minutes = 30/60 = 0.5 degree.
Therefore 30°30′ = 30.5°.
Decimal degrees to DMS require fractional decomposition
The integer portion becomes degrees.
Multiply the fractional degree by 60 for minutes, then multiply the remaining fractional minute by 60 for seconds.
Scenario 21: 12.345° to DMS
Degrees = 12.
0.345 × 60 = 20.7 minutes.
Minutes = 20.
0.7 × 60 = 42 seconds.
Result = 12°20′42″.
If DMS is supported, preserve sign separately
A negative angle applies to the full degrees-minutes-seconds quantity.
Do not make only the degree component negative while treating minutes and seconds as independent signed quantities.
Bearings use direction conventions
A navigation bearing such as N 30° E is not simply the same notation as a standard Cartesian angle of 30°.
Bearing conversion should be treated as a separate feature if added later.
Angles and slopes are related but not identical
Slope can be calculated as tan(θ) for a line angle under the appropriate coordinate convention.
Converting the angle unit does not itself calculate slope.
The Area and Perimeter Calculator will use angles in some geometry modes
Shapes such as sectors, triangles, and regular polygons can involve angle measurements.
Angle Conversion ensures those values are in the unit required by the selected geometry formula.
Sector formulas often require radians
Arc length s = rθ and sector area A = 1/2 r²θ take their simplest form when θ is measured in radians.
Degree inputs must be converted before direct use in those forms.
Scenario 22: 90° sector
90° = π/2 radians.
Using θ = π/2 in A = 1/2 r²θ gives A = πr²/4, one quarter of the circle.
Degree-based sector formulas include conversion explicitly
Sector area can also be written A = (θ°/360°)πr².
Both forms are equivalent when units are handled correctly.
Rotational physics often uses radians
Angular displacement, angular velocity, and angular acceleration frequently use radians in SI-oriented formulas.
Static angle conversion can be the first step before rate calculations.
Angle unit conversion is multiplicative
No nonlinear transformation is needed between these units.
Every conversion is multiplication by a constant factor.
Conversion factors should be reciprocal pairs
If degrees-to-radians uses π/180, radians-to-degrees must use 180/π.
Round-trip tests are useful implementation checks.
Scenario 23: Round-trip conversion
Start with 73°.
Convert to radians and then back to degrees using full precision.
The final value should return to approximately 73°, with only negligible numerical rounding error.
Use full precision internally
Do not convert π to 3.14 before performing the calculation.
Use the runtime’s highest appropriate π constant and round only the displayed result.
The strongest result shows all equivalent units
Even when one target unit is selected, a secondary panel can show degrees, radians, gradians, and turns together.
This makes the relationships among angle systems immediately visible.
Frequently asked questions
How do I convert degrees to radians?
Multiply degrees by π/180.
How do I convert radians to degrees?
Multiply radians by 180/π.
How many radians are 180 degrees?
π radians.
How many radians are 360 degrees?
2π radians.
How many degrees are π radians?
180 degrees.
How many degrees are 2π radians?
360 degrees.
What is 90 degrees in radians?
π/2 radians.
What is 60 degrees in radians?
π/3 radians.
What is 45 degrees in radians?
π/4 radians.
What is 30 degrees in radians?
π/6 radians.
What is π/4 radians in degrees?
45 degrees.
What is π/3 radians in degrees?
60 degrees.
What is π/2 radians in degrees?
90 degrees.
What is a radian?
A radian is the angle subtended when arc length equals the circle’s radius.
Why are there 2π radians in a circle?
A circle’s circumference is 2πr. Dividing that arc length by radius gives 2π radians.
Why are radians used in calculus?
Radians make many trigonometric derivative and integral formulas take their natural simplest forms.
What is a gradian?
A gradian divides a full revolution into 400 units, making a right angle exactly 100 gradians.
How many gradians are in 360 degrees?
400 gradians.
How do I convert degrees to gradians?
Multiply degrees by 10/9.
How do I convert gradians to degrees?
Multiply gradians by 9/10.
What is one turn?
One turn is one complete revolution: 360°, 2π radians, or 400 gradians.
How many degrees are in half a turn?
180 degrees.
How many radians are in half a turn?
π radians.
How many degrees are in a quarter turn?
90 degrees.
How do I convert turns to degrees?
Multiply turns by 360.
How do I convert degrees to turns?
Divide degrees by 360.
How do I convert turns to radians?
Multiply turns by 2π.
Can an angle be more than 360 degrees?
Yes. Angles can represent multiple revolutions.
Can an angle be negative?
Yes. Negative angles commonly represent rotation in the opposite direction from positive angles.
What are coterminal angles?
Angles differing by a whole number of complete revolutions have the same terminal direction.
Are 30 degrees and 390 degrees the same angle?
They are coterminal because they differ by 360 degrees, but their numerical angle measures are different before normalization.
How do I normalize an angle?
Add or subtract complete revolutions until the angle lies in the chosen interval, such as 0° to less than 360°.
Is conversion the same as normalization?
No. Conversion changes units; normalization chooses an equivalent representative angle.
What is DEG mode?
It tells a scientific calculator to interpret trig angles in degrees.
What is RAD mode?
It tells a scientific calculator to interpret trig angles in radians.
Why does sin(30) change between DEG and RAD modes?
The number 30 represents different geometric angles in degrees and radians.
Can I convert degrees, minutes, and seconds?
Yes if DMS mode is supported. One degree contains 60 arcminutes and one arcminute contains 60 arcseconds.
What is 30 degrees 30 minutes in decimal degrees?
30.5 degrees.
Should radian results use pi or decimals?
Exact π-based forms are preferable when available, with decimal approximations shown alongside them.
How accurate is an angle conversion calculator?
The conversion relationships are exact. Decimal results involving π are finite approximations of exact irrational values.
Sources and review
- Angles — OpenStax. Accessed 2026-09-01.
- Unit Circle: Sine and Cosine Functions — OpenStax. Accessed 2026-09-01.
- Trigonometric Functions — NIST Digital Library of Mathematical Functions. Accessed 2026-09-01.
Reviewed 2026-09-01 by Dr Akawak Ejigu, DBA.