Toolarithm

Unit Converter

Degrees to Radians Converter

Exact pi. Clear angle formats.

Convert an angle

Exact pi and angle formats.

Decimal, scientific notation, fraction or proper mixed number. For radian units: pi, -pi/2 or 2*pi/3. For other units, coefficient/pi is supported for exact inverse conversions.

The main conversion preserves every complete rotation. A wrapped representative is reported separately. Symbolic pi remains exact; a typed decimal close to pi is never snapped to a familiar angle.

View results
Precision and report

Decimal/scientific entry allows twenty significant digits, up to eighty characters and written exponents from negative thirty through thirty. Fractions allow one hundred digits per component and two hundred forty characters per field. Decimal certificates describe the pi evaluation model. Symbolic expressions are exact. Decimal values use bounded high-precision pi evaluation and do not create measurement accuracy.

Worked examples

Original angle in radians

θt≈3.14159265359 rad\theta_t\approx3.14159265359\,\mathrm{rad}

The original angle retains its sign and complete rotations. Symbolic expressions preserve pi; the decimal view is rounded separately.

Original angle conversion record

Original degrees

θdeg=180.000000000 ∘\theta_{\mathrm{deg}}=180.000000000\,{}^{\circ}

Original radians

θrad≈3.14159265359 rad\theta_{\mathrm{rad}}\approx3.14159265359\,\mathrm{rad}

Original complete turns

θturn=0.500000000000 turn\theta_{\mathrm{turn}}=0.500000000000\,\mathrm{turn}

Source unit: Degrees. Target unit: Radians. The pi-radian coefficient is multiplied by pi radians; it is not an ordinary-radian decimal. Arcminutes and arcseconds are angular subdivisions, not units of elapsed time.

Exact target expression

θt=(π) rad\theta_t=\left(\pi\right)\,\mathrm{rad}

A rational multiple of pi and a decimal approximation of pi are different inputs. Ordinary radian decimals can produce exact inverse-pi expressions in degrees. No familiar-angle recognition changes the entered number. Numerical evaluation certificates describe a bounded pi model rather than an exact irrational decimal.

Rounded DMS of the original angle

θDMS=(180∘+0′+0.000′′)\theta_{\mathrm{DMS}}=\left(180{}^{\circ}+0{}^{\prime}+0.000{}^{\prime\prime}\right)

One sign applies to the whole degrees, minutes and seconds. Rounding carries through seconds and minutes; it never feeds back into conversion or normalization. DMS based on an ordinary-radian input is a numerical pi evaluation.

Separate wrapped angle

θn∈[0∘,360∘)\theta_n\in[0^{\circ},360^{\circ})

Wrapped degrees

θn,deg=180.000000000 ∘\theta_{n,\mathrm{deg}}=180.000000000\,{}^{\circ}

Wrapped radians

θn,rad≈3.14159265359 rad\theta_{n,\mathrm{rad}}\approx3.14159265359\,\mathrm{rad}

Removed whole rotations: k=0k=0. The range includes its left endpoint and excludes its right endpoint. The exact representative is inside that interval even when its rounded decimal display reaches an endpoint. A negative source uses a floor quotient, not a sign-preserving remainder.

Original angle in every supported unit

All eight rows describe the unwrapped source angle. Pi-radian coefficient values must be multiplied by pi radians. Complete copy and CSV retain every row and its exact symbolic expression even when this table is hidden.

Original angle in eight units
Angle unitOriginal angle
Degrees180.000000000 ∘180.000000000\,{}^{\circ}
Radians≈3.14159265359 rad\approx3.14159265359\,\mathrm{rad}
Pi-radian coefficient1.00000000000 π rad1.00000000000\,\pi\,\mathrm{rad}
Turns / revolutions0.500000000000 turn0.500000000000\,\mathrm{turn}
Gradians / gon200.000000000 gon200.000000000\,\mathrm{gon}
Arcminutes (angle)10800.0000000 ′10800.0000000\,{}^{\prime}
Arcseconds (angle)648000.000000 ′′648000.000000\,{}^{\prime\prime}
Milliradians≈3141.59265359 mrad\approx3141.59265359\,\mathrm{mrad}

Step-by-step angle conversion

  1. 1. Interpret the complete source angle

    The source unit and optional whole-angle DMS sign determine the entered angle. Pi syntax is parsed by a restricted grammar rather than evaluated as code.

    θ=π rad\theta=\pi\,\mathrm{rad}

  2. 2. Apply the exact target definition

    A full revolution defines the conversion factors. Keep symbolic pi until a decimal display is requested; converting does not remove complete rotations.

    θt=π rad\theta_t=\pi\,\mathrm{rad}

  3. 3. Find a separate wrapped representative

    Use a floor quotient and the selected half-open range. The reported whole rotations satisfy the original angle decomposition, including negative sources. Display rounding can reach an endpoint without changing the internal representative.

    θ=2πk+θn,k=0,θn=π rad\theta=2\pi k+\theta_n,\quad k=0,\quad \theta_n=\pi\,\mathrm{rad}
What Can You Create?

Translate original angles, precise notation and angular subdivisions

Move between degree and radian notation

Retain an exact pi coefficient for a stated degree value or symbolic radian input. Compare the exact expression with its rounded decimal view, including negative angles and values containing several complete revolutions.

Read or write compound DMS

Convert a whole-angle sign with degrees, angular minutes and angular seconds into a single quantity. Inspect rounded DMS output with proper carry handling and keep those angular subdivisions separate from elapsed time.

Compare coterminal representatives

Choose a nonnegative full-turn or signed half-turn interval without replacing the original angle. Keep the whole-rotation quotient with the wrapped result so accumulated rotation is not confused with terminal direction.

Degrees and radians formulas with worked examples

Start with one complete revolution

The degree and radian describe the same plane angle with different numerical measures. A full revolution is 360∘360^{\circ} or 2π rad2\pi\,\mathrm{rad}, which fixes both conversion directions. Radians connect an angle to the ratio of circular arc length and radius, θ=s/r\theta=s/r, when both lengths use compatible units. This converter changes the angle representation; it does not infer an arc length, radius or measured direction from a missing geometric context.

θrad=θdegπ180,θdeg=θrad180π\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180},\qquad \theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}

Example: preserve an exact half-turn and quarter-turn

The original half-turn example gives 180∘=π rad180^{\circ}=\pi\,\mathrm{rad}. A quarter-turn gives 90∘=π/2 rad90^{\circ}=\pi/2\,\mathrm{rad}. Entering π/2\pi/2 directly in the radian unit also produces exactly 90∘90^{\circ}. In contrast, an ordinary input of 1 rad1\,\mathrm{rad} gives the exact expression (180/π)∘(180/\pi)^{\circ} and a rounded decimal near 57.2958∘57.2958^{\circ}. The converter does not replace an ordinary decimal with a nearby symbolic angle, even when the decimal display makes them look identical at a low precision.

90π180=π2,1 rad=(180π)∘90\frac{\pi}{180}=\frac{\pi}{2},\qquad 1\,\mathrm{rad}=\left(\frac{180}{\pi}\right)^{\circ}

Example: one sign for the entire compound angle

For a positive angle of twelve degrees, thirty arcminutes and fifteen arcseconds, first convert each subdivision into a degree fraction. The sum is 12+30/60+15/360012+30/60+15/3600 degrees. A negative whole-angle sign negates that entire sum. It does not merely negate the whole-degree component, which would give an incorrect answer for a negative subdegree angle. Components must stay within the stated remainder ranges rather than silently overflowing into a different input interpretation.

θ=σ(d+m60+s3600)∘,σ∈{−1,1}\theta=\sigma\left(d+\frac{m}{60}+\frac{s}{3600}\right)^{\circ},\qquad \sigma\in\{-1,1\}

Example: wrapping retains a separate original angle

For 810∘810^{\circ}, removing two full revolutions leaves 90∘90^{\circ}. The original radian result remains 9π/29\pi/2 rather than becoming π/2\pi/2. In the nonnegative range, −30∘-30^{\circ} uses a floor quotient of negative one and becomes 330∘330^{\circ}. In the signed range the same original angle remains −30∘-30^{\circ}. A half-turn lies on a boundary: the selected signed convention returns −180∘-180^{\circ}, not the excluded positive endpoint. The exact internal representative respects these intervals before presentation rounding.

k=⌊θ2π⌋,θn=θ−2πkfor the nonnegative rangek=\left\lfloor\frac{\theta}{2\pi}\right\rfloor,\quad \theta_n=\theta-2\pi k\quad\text{for the nonnegative range}k=⌊θ2π+12⌋for the signed rangek=\left\lfloor\frac{\theta}{2\pi}+\frac12\right\rfloor\quad\text{for the signed range}

Choose display precision without changing the source

Significant figures control decimal views, while the seconds precision controls only the DMS decomposition. Rounding the total angular seconds before decomposing automatically carries a seconds remainder into minutes and a minutes remainder into degrees. Neither rounded representation becomes the next calculation input. A wrapped angle extremely close to 360∘360^{\circ} can display that endpoint at a low precision even though its internal value is below it. The explanation and exact source make that distinction explicit instead of introducing an arbitrary tolerance or snapping rule.

Why Users Love This Tool

Keep exact notation, decimal evaluation and wrapping distinct

Preserve the mathematical meaning

  • Exact rational coefficients and symbolic pi are retained through compatible unit changes.
  • Ordinary radian decimals are never snapped to special-angle values.
  • Whole-angle DMS signs and half-open wrapping intervals are explicit and independently validated.

Retain a complete conversion record

  • Eight unit rows describe the same unwrapped angle rather than mixing original and wrapped quantities.
  • Separate DMS rounding includes carry and a nonzero-to-zero notice.
  • Text and CSV retain hidden rows, symbolic expressions, evaluation certificates and generated working.
Perfect For

Useful for trigonometry, angular records and rotation comparisons

Students and teachers

Connect exact degree-to-radian fractions to familiar full turns, half-turns and quarter-turns. Compare ordinary radians with pi coefficients, or explain why a negative angle needs floor-based normalization instead of a signed remainder.

Readers of angular DMS records

Translate an explicitly labeled angle with a consistent global sign and valid remainders. This is angle conversion, not a coordinate reference transformation, geodetic datum conversion or determination of an instrument uncertainty.

Readers of rotational quantities

Keep accumulated revolutions visible while optionally comparing the terminal direction. Turns, gradians and milliradians have explicit mathematical definitions, avoiding a silent substitution of an angle for an angular rate or an unspecified military mil convention.

How It Works

How it works in three quick steps.

1

Identify the source angle format

Choose a single angle value or compound degrees, arcminutes and arcseconds. Single values accept decimals, scientific notation, fractions and proper mixed numbers. In radian units, a restricted symbolic pi form is also available. For DMS, enter nonnegative components and choose one sign for the entire angle; minutes and remaining seconds must stay below the next degree or minute.

2

Choose the destination and a separate wrapped range

Select degrees, radians, pi-radian coefficient, turns, gradians, arcminutes, arcseconds or milliradians. The main answer always converts the original angle, including its complete rotations. Optionally choose a nonnegative full-turn or signed half-turn representative. Exact symbolic inputs retain their pi relationship; ordinary decimal radians remain ordinary decimals rather than being snapped to a familiar angle.

3

Review precision, working and saved context

Choose significant figures for decimal views and decimal places for DMS seconds. Inspect the exact expression, original-angle table, separate wrapped values and generated working. Copy the record or LaTeX, download CSV, share validated inputs or print the selected report. Exact swapping exchanges units while preserving the original quantity and its pi coefficient; compound entry becomes a single angle value.

Download & Print

Save the original angle and its notation choices

Copy context or LaTeX

Complete text includes source units, DMS sign when applicable, exact symbolic expressions and all original-angle rows. Mathematical copying supplies proper LaTeX working. Clipboard failures are announced with a downloadable CSV alternative.

Download the complete CSV

The rectangular UTF-8 record retains every row, numerical evaluation certificates, pi bounds, optional wrapped comparison and DMS components. Very long integer or fraction fields should be imported as text when spreadsheet software would otherwise round them.

Print the chosen report

Select the visible exact expression, DMS, original-angle table and calculation steps before printing or saving PDF. Controls and this supporting article are excluded. Long exact coefficients receive a compact-print notice while complete expressions remain available in copied records.

FAQ

Frequently Asked Questions

How do degrees convert to radians and radians to degrees?
A complete revolution is 360∘=2π rad360^{\circ}=2\pi\,\mathrm{rad}. Therefore multiply a degree value by π/180\pi/180 to obtain radians, or multiply a radian value by 180/π180/\pi to obtain degrees. The converter keeps this relationship symbolic before producing a rounded decimal view. For example, 180∘=π rad180^{\circ}=\pi\,\mathrm{rad} exactly. Converting units does not discard complete rotations, change the sign or automatically choose a coterminal representative; those comparisons are reported separately.
What is the difference between symbolic pi and a decimal close to pi?
A symbolic entry such as π/2\pi/2 denotes an exact irrational angle in radians. A typed decimal such as 3.1415926535897933.141592653589793 is an exact finite decimal input, even though it is close to π\pi. Those values are not silently merged. In radian or milliradian input, use forms such as π\pi, −π/2-\pi/2 or 2π/32\pi/3. The separate pi-radian coefficient unit instead expects the numerical coefficient; entering one-half there represents π/2 rad\pi/2\,\mathrm{rad}.
How should a negative DMS angle be entered?
Choose a negative whole-angle sign and enter nonnegative components. The model is θ=−(d+m/60+s/3600)∘\theta=-\left(d+m/60+s/3600\right)^{\circ}, so the sign applies to all three parts. For example, zero degrees and thirty arcminutes under a negative sign represent −0.5∘-0.5^{\circ}. Whole degrees must be integers; whole arcminutes run from zero through fifty-nine, and remaining arcseconds must be below sixty. Complete rotations are allowed in the whole-degree component.
How do the two wrapped angle ranges differ?
The nonnegative range is [0∘,360∘)[0^{\circ},360^{\circ}) and the signed range is [−180∘,180∘)[-180^{\circ},180^{\circ}). Each includes its left endpoint and excludes its right endpoint. Thus 360∘360^{\circ} wraps to 0∘0^{\circ} in the first range, while 180∘180^{\circ} wraps to −180∘-180^{\circ} in the second. Negative inputs use a floor quotient rather than a language-specific signed remainder. The original angle remains visible and retains its complete rotations regardless of this choice.
Can the original angle and wrapped angle have different values?
Yes. They describe the same terminal direction but can differ by complete revolutions. For instance, an original angle of 810∘810^{\circ} converts to 9π/2 rad9\pi/2\,\mathrm{rad}, while its nonnegative representative is 90∘=π/2 rad90^{\circ}=\pi/2\,\mathrm{rad}. The result records the removed whole rotations through θ=2πk+θn\theta=2\pi k+\theta_n. Use the original conversion when accumulated rotation matters. Use the wrapped comparison when a terminal direction or a selected interval is the relevant quantity.
Are arcminutes, arcseconds and gradians time or military units?
Here an arcminute is 1/601/60 of a degree and an arcsecond is 1/36001/3600 of a degree. They are angular subdivisions, not elapsed minutes or seconds. A full revolution contains 400 gon400\,\mathrm{gon}, so a gradian right angle is 100 gon100\,\mathrm{gon}. A milliradian is exactly 0.001 rad0.001\,\mathrm{rad}. It is not automatically one of the differently defined military mil conventions. Match the unit label in the source material before comparing measurements.
What happens when rounding seconds carries into the next minute?
DMS output is rounded as one total angular-second quantity, then decomposed back into degrees, minutes and seconds. That prevents a displayed remainder of sixty seconds or sixty minutes. For example, 12∘+59′+59.9999′′12^{\circ}+59^{\prime}+59.9999^{\prime\prime} rounds to 13∘13^{\circ} at three seconds decimal places. This is a presentation result; the unrounded original angle still drives conversion, exact swapping and normalization. A very small nonzero angle can also round to zero in DMS without being erased internally.
What do the precision controls and export certificates mean?
Decimal displays support three through twelve significant figures, while DMS seconds have an independent zero-through-six decimal-place control. Exact expressions retain rational coefficients and symbolic pi. Decimal evaluation uses a high-precision bounded pi model, so its rational certificate documents that numerical evaluation rather than claiming an exact irrational decimal. Complete copy and CSV retain all unit rows, the original context, wrapped comparisons and generated working, including hidden sections. If pi bounds cannot decide an exceptionally close decimal or DMS rounding boundary, the tool retains the exact expression and reports the uncertainty instead of guessing. These controls do not establish the uncertainty of an angle measurement.
About This Tool

About this Degrees and Radians Converter

The Toolarithm Degrees and Radians Converter treats an angle as a quantity whose representation can change without changing its accumulated rotation. It retains the original half-turn example and supports the familiar degree and radian directions through one clearly labeled source and destination. The expanded unit catalog includes pi-radian coefficients, turns, gradians, arcminutes, arcseconds and milliradians. Every original-angle table row describes the same unwrapped source. An optional wrapped comparison is reported separately, so an angle containing several revolutions is not silently replaced by a smaller terminal direction.

Defined conversion relationships use rational coefficients with symbolic pi retained wherever it belongs. Ordinary radian decimals remain finite decimal inputs and are not promoted to exact familiar angles by a proximity test. Numerical views use high-precision pi bounds, while the exact expression records the underlying mathematical relationship. The floor quotient for ordinary-radian wrapping is checked across those bounds; if it cannot be certified near a boundary, the original conversion remains available with a clear normalization notice. This distinction keeps exact definitions, finite numerical evaluation and display rounding understandable rather than treating them as interchangeable.

Compound entry gives degrees, angular minutes and angular seconds one explicit whole-angle sign. Its valid remainder ranges prevent ambiguous input, including negative angles smaller than a degree. DMS output rounds the complete angular-second quantity before decomposition so carries are handled consistently. Significant figures and seconds decimal places are independent display preferences. Exact swapping retains the original quantity and can use a reciprocal-pi coefficient when that is the correct inverse conversion. Complete copying and CSV retain all rows and hidden working, while share links store validated active settings. The tool converts angle notation; it does not infer a coordinate datum, instrument calibration, measurement uncertainty or angular velocity from the entered number.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Conversion factors are normalized to canonical units and checked in both directions. Exact definitions and offset formulas are preferred over rounded lookup values.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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