Angle Converter
Convert between degrees, radians, and gradians.
How to use Angle Converter
Enter your angle value
Type your numeric angle value into the input field labeled 'Enter Angle'. You can use whole numbers (180) or decimals (45.5).
Select the input unit
Click the dropdown menu next to the input field and choose your current unit: Degrees (°), Radians (rad), or Gradians (gon).
Choose output units
Select which unit(s) you want to convert to using the checkboxes labeled 'Degrees', 'Radians', and 'Gradians' in the output section.
View instant results
Your converted values appear automatically in the results panel below. Copy any result using the copy icon next to each converted value.
Related Tools
Angle converter: degrees, radians, gradians, and more
Angle converter: degrees, radians, gradians, and more
Convert between degrees, radians, gradians, arcminutes, arcseconds, and turns at ToolHQ's angle converter, instant results, all common angle units supported. Free, no account required.
Angles are measured in different units depending on the field. Mathematicians and physicists use radians. Navigators, surveyors, and GPS systems use degrees and DMS (degrees-minutes-seconds). Civil engineers and some European surveying traditions use gradians. Converting between them requires knowing the right formula, or using a converter that handles it instantly.
ToolHQ's angle converter accepts any value in any supported unit and returns the equivalent in all others simultaneously.
Key Takeaways
- Convert degrees, radians, gradians, arcminutes, arcseconds, and turns
- Key formula: multiply degrees by pi/180 to get radians
- Gradians divide a full circle into 400 equal parts, useful in surveying
- DMS notation (47° 30′ 15″) is used in GPS, maps, and astronomy
- No data is stored or transmitted, all calculations run locally in your browser
Angle unit overview
Each angle unit divides the full circle differently:
| Unit | Full circle | Used in |
|---|---|---|
| Degrees (°) | 360° | Everyday use, navigation, geometry |
| Radians (rad) | 2pi (approx. 6.2832) | Mathematics, physics, engineering |
| Gradians / Gon (g) | 400g | Civil engineering, European surveying |
| Turns (tr) | 1 turn | Rotational mechanics, digital graphics |
| Arcminutes (′) | 21,600′ | Navigation, astronomy, GPS precision |
| Arcseconds (″) | 1,296,000″ | Astronomy, precision positioning |
All calculations run locally in your browser, no data is stored or transmitted.
When to use each angle unit
The unit you need depends on your field and application:
Degrees: The most familiar unit. Use degrees for everyday geometry, navigation bearings, roof pitches, and any context where you are communicating with non-specialists.
Radians: The natural unit for mathematics. Most trigonometry formulas (sin, cos, tan) and calculus operations assume radians. Programming languages and spreadsheet functions (sin(), cos()) typically use radians by default. Use radians when working with math or code.
Gradians: A full right angle equals exactly 100 gradians, making gradians convenient for calculations where right angles appear frequently. French civil engineers developed gradians during metrication. They remain common in land surveying in parts of Europe.
DMS (Degrees-Minutes-Seconds): GPS coordinates, nautical charts, and astronomical positions use this format. One degree equals 60 arcminutes, and one arcminute equals 60 arcseconds. A coordinate like 47° 30′ 15″ means 47 degrees, 30 minutes, 15 seconds.
Arcminutes and arcseconds: Used in astronomy for measuring angular distances between stars or the apparent size of objects. Also used in optics and precision engineering.
Mini-story 1: Lukas was writing a physics simulation in Python and kept getting wrong outputs from his trigonometry calculations. The program was calculating the angle of a projectile's trajectory using degrees, but Python's math.sin() function takes radians. A 45-degree launch angle was being passed as 45 to the function, but math.sin(45) gives the sine of 45 radians, not 45 degrees. He entered 45 degrees into ToolHQ's angle converter and got the radian equivalent: 0.7854 rad. Replacing 45 with 0.7854 in his code (or using math.radians(45) to convert inline) immediately fixed the simulation.
How to use the ToolHQ angle converter
The process is immediate:
- Go to the tool. Navigate to ToolHQ's angle converter. No account or sign-up required.
- Enter your angle. Type any numeric value in the input field.
- Select your source unit. Choose the unit you are converting from (degrees, radians, gradians, etc.).
- Read all conversions. The tool returns the equivalent value in all supported units simultaneously.
Conversion formulas
| Conversion | Formula |
|---|---|
| Degrees to radians | radians = degrees x (pi / 180) |
| Radians to degrees | degrees = radians x (180 / pi) |
| Degrees to gradians | gradians = degrees x (10 / 9) |
| Gradians to degrees | degrees = gradians x (9 / 10) |
| Degrees to arcminutes | arcminutes = degrees x 60 |
| Arcminutes to degrees | degrees = arcminutes / 60 |
| Degrees to arcseconds | arcseconds = degrees x 3600 |
| Degrees to turns | turns = degrees / 360 |
Why pi/180? A full circle contains 360 degrees and also exactly 2pi radians. So 1 degree = 2pi/360 = pi/180 radians. This is why multiplying by pi/180 converts degrees to radians, and multiplying by 180/pi converts back.
Common angles reference table
| Angle (degrees) | Radians | Gradians | Turns |
|---|---|---|---|
| 0° | 0 rad | 0g | 0 tr |
| 30° | pi/6 (0.5236) | 33.33g | 1/12 tr |
| 45° | pi/4 (0.7854) | 50g | 1/8 tr |
| 60° | pi/3 (1.0472) | 66.67g | 1/6 tr |
| 90° | pi/2 (1.5708) | 100g | 1/4 tr |
| 120° | 2pi/3 (2.0944) | 133.33g | 1/3 tr |
| 180° | pi (3.1416) | 200g | 1/2 tr |
| 270° | 3pi/2 (4.7124) | 300g | 3/4 tr |
| 360° | 2pi (6.2832) | 400g | 1 tr |
Note on gradians: A 90-degree right angle equals exactly 100 gradians, which is why gradians simplify calculations involving right angles and quadrants.
For converting other unit types, ToolHQ's unit converter handles length, weight, temperature, and speed conversions across all major unit systems.
Mini-story 2: Sofia was a land surveyor in training and encountered gradians for the first time when reviewing notes from a senior colleague. The colleague had recorded a bearing as 73.5g and Sofia needed to report it in degrees for the project's GIS software. She entered 73.5 gradians into ToolHQ's angle converter and got 66.15 degrees immediately. The experience also taught her that 100g equals a right angle, which helped her understand why the colleague's field notes were organized the way they were.
Frequently asked questions
How do I convert degrees to radians?
Multiply the degrees by pi/180. Example: 90 degrees x (3.14159/180) = 1.5708 radians. Alternatively, use ToolHQ's angle converter and enter the value in degrees.
What is a gradian and why does it exist?
A gradian (also called a gon) divides a full circle into 400 equal parts, so a right angle equals exactly 100 gradians. This was developed as part of metric system standardization in France and remains common in civil engineering and European surveying because right angles appear as round numbers.
What is DMS notation?
DMS stands for Degrees-Minutes-Seconds. One degree equals 60 arcminutes, and one arcminute equals 60 arcseconds. The format 47° 30′ 15″ means 47 degrees, 30 arcminutes, and 15 arcseconds. GPS coordinates are often expressed this way.
Why do programming languages default to radians?
Mathematical functions like sin() and cos() are defined in terms of radians in calculus and the underlying mathematics. Defaulting to radians avoids a constant conversion step in the mathematical operations that generate these functions. If you pass degrees, multiply by pi/180 first, or use a dedicated conversion function.
Can I convert negative angles?
Yes. Negative angles are valid and represent rotation in the opposite direction (clockwise vs. counterclockwise, depending on convention). Enter a negative value and the converter returns the negative equivalent in all other units. For example, -90 degrees equals -pi/2 radians.
Is the angle converter free?
Yes. ToolHQ's angle converter is completely free, with no account, no sign-up, and no usage limits.
The short version
Degrees are for everyday use. Radians are for mathematics and programming. Gradians are for surveying and civil engineering. DMS (degrees-minutes-seconds) is for navigation and GPS. The formulas are simple once you understand the rationale: degrees times pi/180 gives radians because a full circle is both 360 degrees and 2*pi radians. ToolHQ's angle converter handles all six units simultaneously, returning every equivalent value from a single input. It is free, instant, and requires no account.
For related conversions, ToolHQ's unit converter covers length, weight, temperature, speed, and other physical quantities, and the area converter converts between square meters, acres, hectares, and other area units. Explore more converter tools at ToolHQ.