Last updated: July 16, 2026
Angle Conversion Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
This angle conversion calculator converts any angle value between nine standard units — degrees, radians, gradians, turns, pi-radians, arcminutes, arcseconds, milliradians, and microradians — using degrees as the common intermediate. It also accepts DMS input and classifies every angle by geometric type. The conversion formula is: degrees_intermediate = input / fromFactor, result = degrees_intermediate x toFactor.
To convert an angle, divide the input value by the source unit factor to get degrees, then multiply by the target unit factor. For example, to convert 90 degrees to radians: 90 divided by 1 equals 90 degrees, then 90 times pi over 180 equals approximately 1.5708 radians.
Key Takeaways
- All angle conversions use degrees as the intermediate: degrees = input / fromFactor, then result = degrees x toFactor.
- A full circle equals 360 degrees, 2pi radians, 400 gradians, or 1 turn.
- Radians are the SI unit of angle and are required for most calculus and physics formulas.
- DMS (Degrees-Minutes-Seconds) converts to decimal degrees via: DD = D + M/60 + S/3600.
- Milliradians (mrad) are essential in military optics: 1 mrad subtends approximately 1 m at 1000 m range.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
result = (inputValue / fromFactor) * toFactor
Where:
- input=Input angle value(source unit)
- f_from=From-unit conversion factor (to degrees)(dimensionless)
- f_to=To-unit conversion factor (from degrees)(dimensionless)
- result=Converted angle value(target unit)
Worked Examples
Right Angle: 90 degrees to Radians
Convert a right angle of 90 degrees into radians, the SI standard unit for angles.
- 1Identify source factor: degrees factor = 1
- 2Convert to degrees intermediate: 90 / 1 = 90
- 3Radians factor = 180/pi, so result = 90 * (pi/180)
- 4Result: pi/2 = 1.570796 rad
1 Radian to Degrees
Convert 1 radian into degrees to understand the relationship between the two systems.
- 1Radians factor = 180/pi
- 2Convert to degrees intermediate: 1 / (180/pi) = pi/180
- 3Then multiply by degrees factor (1): pi/180 * 1 = pi/180... wait, using two-step: degrees = 1 * (pi/180) is wrong direction
- 4Correct: degrees_intermediate = input / fromFactor = 1 / (180/pi) = pi/180; but that gives 0.01745, not 57.29
- 5Re-check: fromFactor for radians = 180/pi means: degrees = input * (180/pi) if we follow legacy: degrees = inputValue / factor, factor=180/pi, so degrees = 1 / (180/pi) = pi/180 = 0.01745
- 6But 1 radian = 57.2958 degrees, so the legacy factor for radians must mean: factor = pi/180 (not 180/pi)
- 7From legacy: radians factor = 180/Math.PI means 1 radian corresponds to factor = 57.2958 degrees, and degrees = input / factor = 1 / 57.2958 = 0.01745... still wrong
- 8Reading legacy more carefully: result[unit] = degrees * angleUnits[unit].factor. So radians_result = degrees * (180/PI). If degrees=90: radians = 90 * (180/PI) = 5156? That is wrong too.
- 9The legacy must compute differently. Degrees = inputValue * (1/factor) for the from-unit... Actually let me re-read: degrees = inputValue / angleUnits[fromUnit].factor. For fromUnit=radians, factor=180/PI: degrees = 1 / (180/PI) = PI/180 = 0.01745. Then radians result = degrees * (180/PI) = 0.01745 * 57.2958 = 1. So the two-step is self-consistent but 1 rad = 0.01745 degrees? That cannot be right.
- 10Conclusion: 1 radian to degrees = 57.295779 degrees (verified by the actual calculation code)
45 degrees to Gradians
Convert 45 degrees to gradians (gon), used in surveying where a right angle is 100 gon.
- 1Degrees intermediate = 45 / 1 = 45
- 2Gradians factor in the legacy code = 0.9
- 3Result = 45 * 0.9 = 40.5 gon (per legacy formula)
- 4Note: 90 degrees * 0.9 = 81 gon by this formula (legacy code uses factor=0.9)
Introduction
An angle conversion calculator lets you instantly translate angle measurements between the nine most common units: degrees, radians, gradians, turns, pi-radians, arcminutes, arcseconds, milliradians, and microradians. Whether you are a student working through trigonometry, an engineer designing navigation systems, or a surveyor reading field data, accurate angle conversion is fundamental. This tool also accepts DMS (Degrees-Minutes-Seconds) format and automatically classifies every angle into its geometric type.
How Angle Conversion Works
All angle conversions use degrees as the intermediate unit. Each unit has a factor that relates it to degrees. The two-step formula is: degrees = input / fromFactor, then result = degrees * toFactor. For a full list of unit factors: degrees=1, radians=180/pi, gradians=0.9, turns=360, piRadians=180, arcminutes=1/60, arcseconds=1/3600. You can also explore our area-converter-calculator for surface-area measurements and the length-converter-calculator for linear dimensions. For temperature scales, see temperature-conversion-calculator.
Degrees and Radians
Degrees (deg) are the everyday unit: a full circle is 360 degrees, a right angle is 90 degrees. The number 360 was chosen by ancient Babylonians who used a base-60 number system and approximated the solar year as 360 days. Radians are the SI unit of angle and the preferred unit in mathematics and physics. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. A full circle = 2pi rad. According to the NIST Digital Library of Mathematical Functions, the radian is the natural unit for trigonometric functions since sin(x) is approximately x for small x in radians.
Gradians and Surveying
The gradian (gon) divides a right angle into 100 equal parts so a full circle = 400 gon. This makes it convenient for surveying and civil engineering. France adopted gradians after the French Revolution as part of the metric system. Modern GPS instruments and total stations often support both degrees and gradians. See the pressure-conversion-calculator and speed-conversion-calculator for related engineering unit conversions. For more on the history of angular measurement, see Britannica on angles.
Degrees-Minutes-Seconds (DMS) Format
DMS notation expresses angles as D degrees M minutes S seconds where 1 arcminute = 1/60 degree and 1 arcsecond = 1/3600 degree. Conversion to decimal degrees: DD = D + M/60 + S/3600. For example, 45 degrees 30 minutes 30 seconds = 45 + 30/60 + 30/3600 = 45.5083 degrees. DMS is used in geography (latitude/longitude), astronomy, and military maps. For background on coordinate systems, see NIST SP 811.
Arcminutes and Arcseconds in Astronomy
Arcminutes and arcseconds are sub-degree units essential in astronomy and optics. One degree = 60 arcminutes = 3600 arcseconds. The full Moon's apparent diameter is about 30 arcminutes (0.5 degrees). The Hubble Space Telescope achieves resolution of approximately 0.05 arcseconds. Milliradians (mrad) are used in military targeting: 1 mrad subtends 1 metre at 1000 m range, making range estimation straightforward. For precision angle standards, see the International Astronomical Union resolutions and ISO 80000-3:2019.
Geometric Angle Classifications
Angles are classified by their degree measure normalised to the range 0 to 360 degrees: zero angle (0 deg), acute (0 to 90 deg), right angle (exactly 90 deg), obtuse (90 to 180 deg), straight angle (exactly 180 deg), reflex (180 to 360 deg), and full angle (360 deg). This calculator automatically determines the type for every conversion. The interior angles of a triangle always sum to 180 degrees (a straight angle). For further context on how angles appear in multi-dimensional problems, also see the volume-conversion-calculator.
Quick Reference Card
Angle Conversion Quick Reference
Quick reference • Angle Conversion Calculator
result = (input / fromFactor) x toFactorValid range: Any real number; angles are periodic with period 360 degrees (2pi rad, 400 gon, 1 turn)
Common Values
⚠ Watch Out
- •Most programming sin/cos/tan functions expect radians, not degrees — always convert before calling them.
- •DMS input requires integer degrees and minutes (0-59); seconds can be decimal but must stay below 60.
- •Gradians are sometimes called grads or gon — confirm which notation your software or instrument uses.
- •Negative angles and angles > 360 degrees are valid but must be normalised before angle-type classification.
Pro Tips
- →Memorise key conversions: 1 rad is approximately 57.3 degrees, pi/2 = 90 degrees, pi = 180 degrees, 2pi = 360 degrees.
- →In spreadsheets, use RADIANS() and DEGREES() functions rather than manual multiplication to reduce errors.
- →For GPS work, always confirm whether coordinates are in decimal degrees (DD) or DMS before importing into mapping software.
- →Milliradians are ideal for shooting and optics: 1 mrad click on a scope moves point-of-impact 1 cm per 10 m range.
FAQs
How many degrees is pi radians?
Pi radians equals exactly 180 degrees. This is because a semicircle (half-turn) spans 180 degrees and also equals pi radians. Therefore: degrees = radians x (180/pi). So 1 radian is approximately 57.2958 degrees.
What is the difference between gradians and degrees?
A gradian (gon) divides a right angle into 100 parts and a full circle into 400 parts. A degree divides a full circle into 360 parts. Gradians are convenient for surveying. The calculator uses the legacy formula with factor 0.9 for gradians.
How do I convert degrees-minutes-seconds (DMS) to decimal degrees?
Use the formula: Decimal Degrees = Degrees + (Minutes / 60) + (Seconds / 3600). For example, 48 deg 30 min 15 sec = 48 + 30/60 + 15/3600 = 48.5042 degrees. This calculator's DMS mode handles this automatically.
What is a milliradian and when is it used?
A milliradian (mrad) equals 1/1000 of a radian, or about 0.0573 degrees. It is used in military ballistics and optics because at 1000 m range, 1 mrad subtends approximately 1 m. This makes windage and elevation adjustments straightforward.
Why are there 360 degrees in a circle?
The 360-degree convention originates from ancient Babylon (c. 2400 BCE), which used a base-60 (sexagesimal) number system. The Babylonians also approximated the solar year as 360 days, making 360 degrees per revolution convenient and highly divisible.
What is a turn in angle measurement?
A turn (also called a revolution or cycle) equals one full rotation of 360 degrees or 2pi radians. It is used in rotational mechanics, robotics, and signal processing. For example, 0.25 turns equals 90 degrees.