Angle Conversion Calculator

Convert any angle between degrees, radians, gradians, turns and π radians.

Supports degrees-minutes-seconds input and an exact π-radian fraction readout.

Updated August 16, 2026
Frank Zhao - Creator
CreatorFrank Zhao
Loading calculator…

Introduction

An angle is the same physical rotation no matter how you write it, but the way you write it depends on where it comes from. Math and physics usually expect radians, maps and instruments use degrees or degrees-minutes-seconds, surveying and some engineering use gradians (gon), and a full rotation is sometimes simplest as a turn. This calculator accepts any of them and instantly shows the same angle in every other unit.

Clean, exact degrees-minutes-seconds

Type a whole degree value like 201∘201^{\circ} and the DMS field reads back a clean 201∘ 0′ 0"201^{\circ}\,0'\,0". Because the calculator works in exact arcseconds internally, you never see the tiny leftover like 1×10−581\times 10^{-58} seconds that some converters show for a plain integer angle — the reading is exact, not a rounded-off approximation.

The π-radian fraction is a real input, not just a read-out

The π rad fraction field shows the angle as a fraction of π\pi radians — 90∘90^{\circ} appears as 12\frac{1}{2}. Unlike a plain display line, this field is editable: type 22 and 33 into its two boxes and the whole calculator re-solves to 120∘120^{\circ}. You can read a π fraction or drive the calculation with one.

Students checking homework, engineers converting a CAD angle, programmers feeding a trig function, and anyone working across degrees and radians will find every field updates together, so a single angle is never re-typed into a different unit by hand.

How to use / quick start

  1. 1Type the angle you already have into any field — Degrees, Deg/min/sec, Radians, Gradians, Turns, π radians or the two-box π rad fraction.
  2. 2Every other field updates instantly. Read whichever form your map, spreadsheet, code or formula needs and copy it.
  3. 3Correct any single value later — because every field is editable, the whole set re-solves around the value you changed. You never re-enter the full angle.

Worked example — read 120∘120^{\circ} in every unit

Type 120120 into the Degrees field. The calculator fills the rest from the same rotation:

120∘120^{\circ}==2π3 rad\frac{2\pi}{3}\ \mathrm{rad}==133.3‾ gon133.\overline{3}\ \mathrm{gon}==13 turn\frac{1}{3}\ \mathrm{turn}

So the fields show 2.0944 rad2.0944\ \mathrm{rad}, 133.333 gon133.333\ \mathrm{gon}, 0.333333 turn0.333333\ \mathrm{turn}, 0.666667 π rad0.666667\ \pi\ \mathrm{rad}, DMS 120∘ 0′ 0"120^{\circ}\,0'\,0" and the π fraction 23\frac{2}{3}.

Worked example — drive the calculation from a π fraction

Suppose you want the angle that is 12\frac{1}{2} of π\pi radians. Type 11 into the numerator box and 22 into the denominator box of π rad fraction:

nd×π rad\frac{n}{d}\times \pi\ \mathrm{rad}==12×π rad\frac{1}{2}\times \pi\ \mathrm{rad}==90∘90^{\circ}

The Degrees field reads 9090. Enter the fraction any way you like — 23\frac{2}{3} gives 120∘120^{\circ}, and a value like 1.8∘1.8^{\circ} is even shown as the exact fraction 1100\frac{1}{100} so the boxes never hold a decimal numerator.

Real-world examples

Map work: a DMS bearing into decimal degrees

A map labels a bearing as 47∘ 23′ 31.2"47^{\circ}\,23'\,31.2", but the spreadsheet needs a single decimal number. Type the three parts into Deg/min/sec and read Degrees:

DD\mathrm{DD}==4747++2360\frac{23}{60}++31.23600\frac{31.2}{3600}==47+0.3833+0.008747 + 0.3833 + 0.0087==47.392∘47.392^{\circ}

Paste 47.39247.392 wherever a single decimal value is expected, and the DMS box keeps showing the original bearing exactly as written.

Trig and physics: write the angle as a clean π fraction

A physics problem wants the angle in radians as an exact multiple of π\pi, such as 5π6\frac{5\pi}{6}. In π radians, type 56\frac{5}{6} (or 0.8333...0.8333...), and π rad fraction shows the reduced integer fraction 56\frac{5}{6} while Degrees reads 150150.

150∘150^{\circ}==150180 π rad\frac{150}{180}\ \pi\ \mathrm{rad}==56 π rad\frac{5}{6}\ \pi\ \mathrm{rad}

The reduced integer fraction is exactly what shows in π rad fraction — no decimal numerator, no clutter.

Calculation method

Degrees and radians

A full circle is 360∘=2π rad360^{\circ} = 2\pi\ \mathrm{rad}, so the two conversions are:

rad\mathrm{rad}==deg\mathrm{deg}×\timesπ180\frac{\pi}{180}
deg\mathrm{deg}==rad\mathrm{rad}×\times180π\frac{180}{\pi}

The same pattern links the rest: gradians use 200200 per half-turn, turns count full rotations, and ×π rad\times \pi\ \mathrm{rad} counts in multiples of π\pi. Between any two units:

AtargetA_{\mathrm{target}}==AsourceA_{\mathrm{source}}×\timesfsourceftarget\frac{f_{\mathrm{source}}}{f_{\mathrm{target}}}

Degrees, minutes and seconds

Minutes and seconds are base-60, the same rule as time: 1∘=60′=3600"1^{\circ} = 60' = 3600". A DMS value becomes a single decimal number as:

DD\mathrm{DD}==DD++M60\frac{M}{60}++S3600\frac{S}{3600}

The calculator keeps every angle in exact arcseconds internally, so a whole-degree input like 201∘201^{\circ} stays exactly 201∘ 0′ 0"201^{\circ}\,0'\,0" with no residual noise — that is why you never see a stray 1×10−581\times 10^{-58} second in the seconds box.

The π-radian fraction field

The field stores the angle as a fraction of a half-turn, because π rad=180∘\pi\ \mathrm{rad} = 180^{\circ}:

A=nd×π radA = \frac{n}{d} \times \pi\ \mathrm{rad}==nd×180∘\frac{n}{d} \times 180^{\circ}

The calculator always shows a reduced fraction with integer numerator and denominator: a finite input like 1.8∘1.8^{\circ} becomes 1100\frac{1}{100}. An angle that cannot be an exact integer fraction — for example 1 rad1\ \mathrm{rad} — is shown as a best integer approximation like 113355\frac{113}{355} with an ≈\approx toggle that reveals the full decimal value.

Tips & best practices

  • Trust a clean DMS reading for whole degrees

    When you type an integer degree value, the degrees-minutes-seconds boxes show whole numbers with zero seconds — 201∘201^{\circ} reads 201∘ 0′ 0"201^{\circ}\,0'\,0". If another converter ever shows a tiny second like 1×10−581\times 10^{-58}, that is arithmetic noise, not a real fraction of a second.

  • Use π rad fraction as a two-way tool

    The fraction boxes are editable inputs. To find the angle that is 56π\frac{5}{6}\pi, just type 55 and 66 — the calculator solves for 150∘150^{\circ}.

  • Understand the ≈ on a π fraction

    An ≈\approx toggle on the fraction means the angle cannot be an exact integer fraction of π\pi (true for most radian values). 1 rad≈1133551\ \mathrm{rad} \approx \frac{113}{355} is the best integer approximation — click the toggle to see the full decimal.

  • Don't mix degrees and radians

    Mixing the two is the most common angle mistake. If a result that should be near 30∘30^{\circ} shows up as about 0.520.52, the source value was almost certainly radians — switch that field to Radians and re-check.

Frequently asked questions

Why does DMS show 201∘ 0′ 0"201^{\circ}\,0'\,0" and not a tiny second value?

Because the calculator works in exact arcseconds, a whole-degree input stays exactly 201∘201^{\circ} with zero minutes and seconds. The tiny 1×10−581\times 10^{-58} second some converters show is just rounding noise from converting through radians; this calculator avoids it.

Can I type into the π rad fraction field?

Yes. The numerator and denominator are real inputs. Type 22 and 33 to get 120∘120^{\circ}, or 11 and 22 for 90∘90^{\circ}.

What does the ≈\approx toggle on the fraction mean?

It means the angle cannot be written as an exact integer fraction of π\pi, which is true for most radian values. The shown integer fraction (for example 113355\frac{113}{355} for 1 rad1\ \mathrm{rad}) is the best approximation; the toggle reveals the full decimal value.

Why did 1.8∘1.8^{\circ} come back as 1100\frac{1}{100}?

The fraction is always reduced with integer numerator and denominator: 1.8180=181800=1100\frac{1.8}{180} = \frac{18}{1800} = \frac{1}{100}. A value like 33.8688∘33.8688^{\circ} becomes 5883125\frac{588}{3125}, so the boxes never hold a decimal numerator.

Limitations

  • This calculator changes how an angle is written; it does not measure angles or add accuracy beyond the numbers you type. The result is only as precise as the input.
  • Most radian values cannot be written as an exact integer fraction of π\pi, so the π fraction uses a best integer approximation (marked with ≈\approx) in those cases.
  • Negative angles and angles beyond a full turn are kept exactly as typed. If your application expects a bearing in a fixed 0∘0^{\circ}–360∘360^{\circ} range, you may need to normalize the value yourself.

Related Calculators

Scale Calculator

Calculate scale factors and convert scaled or real dimensions for length, area, and volume.

Try Now

Roman Numeral Converter

Convert an Arabic number up to 3,999,999 into Roman numerals or read a Roman numeral back into a number. Both directions run in your browser.

Try Now

Trajectory Calculator

Have a look at the flight path of the object with this trajectory calculator.

Try Now

Projectile Range Calculator

Calculate the horizontal range of a projectile based on velocity, angle, and initial height. Supports bidirectional calculation with multiple unit systems.

Try Now

Projectile Motion Calculator

Calculate projectile trajectory parameters including launch velocity, angle, distance, maximum height, and flight time with bidirectional solving.

Try Now

Maximum Height Calculator – Projectile Motion

Use this maximum height calculator to figure out what is the maximum vertical position of an object in projectile motion.

Try Now
Angle Conversion Calculator – Degrees, Radians, Gradians & Turns