Degrees to Radians Converter
Convert degrees to radians or radians to degrees with the exact identity 180° = π rad. Enter signed decimals, fractions, degree-minute-second notation or symbolic π forms, then compare the exact expression with a controlled decimal approximation.
- Exact symbolic π results
- DMS and π-expression input
- Signed and multi-turn angles
- Round only the final decimal
Convert Degrees and Radians
Choose a direction, enter one angle, and set the displayed precision. The original angle stays unchanged. A coterminal angle in the interval from 0° up to, but not including, 360° appears only as a separate reference.
180° converted with the exact relation 180° = π rad.
Calculation Steps
- Use the exact identity 180° = π rad.
- Multiply 180° by π/180 to obtain exactly π rad.
- Use the stored 1,500-decimal-digit working approximation of π only for the decimal comparison.
- Round the final numerical comparison once to 6 decimal places: approximately 3.141593 rad.
- Preserve the full signed angle; show 180° | π rad only as a separate coterminal reference.
The symbolic value π rad is exact. The numerical result uses a 1,500-decimal-digit working approximation of π and 6 decimal places with nearest rounding, ties to even. The converter preserves 180° rather than replacing it with a normalized angle.
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How to Use This Degrees to Radians Converter
Allow about two minutes for one conversion, including a quick check of the input notation and reporting precision.
- Select the direction. Choose degrees to radians or radians to degrees. The formula and accepted notation update with the direction.
- Enter one angle. For degrees, use a signed decimal, scientific notation, fraction, mixed number or DMS form. For radians, use a number or a supported expression containing pi.
- Check the notation. DMS minutes and seconds must each be less than 60. A pi expression may use forms such as pi, -pi/2, 3pi/4 or 2*pi.
- Choose display precision. Select decimal places or significant figures, enter the required count, and choose how exact halfway cases are rounded.
- Press Convert. Review the rounded numerical result, exact symbolic result, equivalent angle measures, DMS form and calculation steps.
- Copy, swap or reset. Copy the labelled result, swap directions without reusing the rounded display, or restore the default 180-degree example.
The converter accepts zero, negative angles and angles covering many turns. It does not force the input into a single revolution. The separate coterminal reference uses the interval from 0° inclusive to 360° exclusive, so an input of 450° remains 450° while the reference shows 90°.
Degrees and Radians Conversion Formulas
The radian, symbol rad, is the coherent SI unit for plane angle. The degree is a non-SI unit accepted for use with the SI.
A complete revolution is exactly 360 degrees and exactly 2π radians. Dividing both sides by two gives 180° = π rad. This identity produces both conversion formulas.
For degree input, the calculator first reduces the rational coefficient multiplying π. For example, 45/180 reduces to 1/4, so 45° equals exactly π/4 rad. The decimal 0.785398... is useful for numerical work, but it remains an approximation because π has no terminating or repeating decimal expansion.
For a radian input written as a rational multiple of π, the reverse result can also be exact. The input 3π/2 rad gives 3/2 × 180° = 270°. A plain input of 1 rad has the exact symbolic degree form 180/π°, while its decimal value is approximately 57.2957795°.
Accepted Angle Notation
Decimal and scientific notation work in both directions. Examples include 90, -22.5, 1e3 and 2.5e-4. Simple fractions such as 1/3 and mixed numbers such as -1 1/2 are stored as ratios of integers rather than binary floating-point values.
Degree-minute-second input
Degree input also accepts colon notation such as 30:15:50 and labelled notation such as 30° 15′ 50″. One leading minus sign applies to the entire angle, including a negative-zero-degree form such as -0° 30′ 0″. Minutes and seconds must stay from 0 inclusive to 60 exclusive.
One arcminute is exactly 1/60 degree. One arcsecond is exactly 1/60 arcminute, or 1/3600 degree. DMS appears as a secondary result because decimal degrees are usually easier to calculate with.
Pi-expression input
Radian input recognizes π or the ASCII word pi. Supported examples include pi, -pi, pi/6, 3pi/4, 0.5pi and 2*pi. The grammar permits one pi term and an optional divisor. It rejects formulas with additions, functions, repeated pi symbols or a zero denominator. The calculator never evaluates arbitrary code.
Worked Degrees and Radians Examples
Example 1: Convert 30 degrees to radians
Multiply 30 by π/180 and reduce 30/180 to 1/6. The exact result is π/6 rad. Its decimal approximation is about 0.5235987756 rad.
Example 2: Convert -135 degrees to radians
The sign stays with the angle. Reduce -135/180 to -3/4. The exact result is -3π/4 rad. The calculator does not replace it with the positive coterminal angle 5π/4 rad.
Example 3: Convert 3pi/2 radians to degrees
The pi factors cancel symbolically. Multiply 3/2 by 180 to obtain exactly 270 degrees.
Example 4: Convert 1 radian to degrees
Multiply 1 by 180/π. The exact symbolic result is 180/π°. Because the decimal contains the irrational constant π, report it with an approximation sign.
Example 5: Convert 30 degrees 15 minutes 50 seconds
Convert the minutes and seconds to degree fractions. The input equals 30 + 15/60 + 50/3600, or 10895/360 degrees, which reduces to exactly 2179/72 degrees. Its exact radian result is 2179π/12960 rad.
Common Degrees to Radians Table
The symbolic values in this table are exact. Decimal values are rounded reference values and should carry an approximation sign when used as equality statements.
| Degrees | Exact radians | Decimal radians | Turns |
|---|---|---|---|
| 0° | 0 rad | 0 | 0 |
| 15° | π/12 rad | ≈ 0.261799 | 1/24 |
| 30° | π/6 rad | ≈ 0.523599 | 1/12 |
| 45° | π/4 rad | ≈ 0.785398 | 1/8 |
| 60° | π/3 rad | ≈ 1.047198 | 1/6 |
| 90° | π/2 rad | ≈ 1.570796 | 1/4 |
| 120° | 2π/3 rad | ≈ 2.094395 | 1/3 |
| 135° | 3π/4 rad | ≈ 2.356194 | 3/8 |
| 150° | 5π/6 rad | ≈ 2.617994 | 5/12 |
| 180° | π rad | ≈ 3.141593 | 1/2 |
| 270° | 3π/2 rad | ≈ 4.712389 | 3/4 |
| 360° | 2π rad | ≈ 6.283185 | 1 |
Full Angles Versus Coterminal Angles
Conversion and normalization answer different questions. Conversion changes the unit while preserving the complete signed angle. Normalization removes whole turns to find a coterminal direction. For example, 810° converts to exactly 9π/2 rad because it contains two full turns plus 90°. Its normalized reference is 90°, or π/2 rad.
This calculator never normalizes the primary result. It calculates the reference from the unrounded internal value and places it in a separate card. The selected interval is [0°, 360°), meaning 0° is included and 360° is represented as 0°. A negative input such as -90° therefore has a 270° coterminal reference while its converted result remains -π/2 rad.
Do not use a rounded decimal to decide whether an angle lies on the 0°/360° boundary. A value slightly below 360° must remain slightly below 360°, even if a short display rounds it to 360.000°.
Exact Values, Pi and Rounding
The calculator stores entered decimals, fractions and DMS components as reduced ratios of integers. Degree-to-radian conversion keeps the rational coefficient of π exact. A radian expression containing π also stays symbolic until π cancels or a decimal comparison is requested.
For numerical results involving π, the engine uses a 1,500-decimal-digit working approximation. This is much longer than ordinary JavaScript number precision and supports the page's display controls, but it is still an approximation of an irrational number. The exact symbolic card remains the authoritative mathematical form.
Choose decimal places when a fixed number of digits after the decimal point is required. Choose significant figures when the overall precision matters across large or small magnitudes. Nearest, ties to even avoids a systematic upward bias across many exact halfway cases. Nearest, ties away from zero is also available when a reporting rule requires it.
The tool rounds once at the final display. If a selected fixed-decimal setting would turn a nonzero result into zero, it uses a six-significant-figure fallback and states why. Extra displayed digits do not improve the accuracy of a measured input.
Common Degree and Radian Mistakes
- Using the factor backwards. Degrees to radians multiplies by π/180. Radians to degrees multiplies by 180/π.
- Dropping pi from an exact answer. Thirty degrees is π/6 rad, not 1/6 rad.
- Calling a rounded decimal exact. The symbolic relation may be exact while its nonzero decimal expansion is approximate.
- Treating radians as degrees in trigonometry. Check the angle mode expected by your calculator, graphing tool or programming function.
- Silently normalizing an angle. A multi-turn or negative angle contains information that a coterminal direction alone does not preserve.
- Applying a sign to degrees only in DMS. One leading sign applies to the whole DMS angle, including minutes and seconds.
- Allowing 60 minutes or 60 seconds. Carry 60 seconds into one minute and 60 minutes into one degree before entering DMS.
Related Calculators
Frequently Asked Questions
These answers cover the exact relationship, accepted notation, signs, normalization and decimal precision.
How do I convert degrees to radians?
Multiply the degree value by pi and divide by 180. For example, 60 degrees times pi over 180 reduces to exactly pi over 3 radians.
How do I convert radians to degrees?
Multiply the radian value by 180 and divide by pi. A symbolic input such as pi over 4 radians converts exactly to 45 degrees.
Why is 180 degrees equal to pi radians?
A full revolution is 360 degrees and 2 pi radians. Dividing both measures by two gives the exact identity 180 degrees equals pi radians.
Is a decimal radian result exact?
Zero can be exact, but a nonzero rational multiple of pi has an irrational decimal expansion. Keep the symbolic pi form for an exact result and label the rounded decimal as approximate.
Can I enter pi in the radians box?
Yes. Supported forms include pi, -pi, pi/6, 3pi/4, 0.5pi and 2*pi. The converter also recognizes the pi symbol and rejects arbitrary formulas.
Can I enter degrees, minutes and seconds?
Yes. Use colon notation such as 30:15:50 or labelled notation such as 30 degrees 15 minutes 50 seconds. Minutes and seconds must each be less than 60.
Does the converter accept negative angles?
Yes. The sign applies to the whole angle. Negative angles and values beyond one turn remain unchanged in the primary conversion result.
Does the calculator reduce every angle to 0 through 360 degrees?
No. It preserves the complete input angle. A separate coterminal reference is shown in the interval from 0 degrees inclusive to 360 degrees exclusive.
How many degrees are in one radian?
One radian equals exactly 180 over pi degrees, which is approximately 57.2957795131 degrees.
How many radians are in one degree?
One degree equals exactly pi over 180 radians, which is approximately 0.0174532925199 radian.
Method and Source Basis
BIPM SI Brochure, 9th edition defines the radian and its plane-angle role. NIST SP 330, section 5.4.8 states the exact relation 360° = 2π rad. NIST SP 811 Appendix B provides converted-value rounding guidance. OpenStax Precalculus 2e, section 5.1 explains the degree-radian proportion, negative angles and coterminal angles.
Calculation note: This tool converts mathematical angle measures. It does not measure a physical angle, establish instrument calibration, determine surveying or navigation conventions, select a programming-language angle mode, or certify an engineering result. Confirm the required notation, interval and precision for your use.