Angle Converter (All-in-One)
Convert between degrees, radians, and grads instantly
Converted Values
Degree (°): 90.00
Radian (rad): 1.57
Gradian (gon): 100.00
Formula
Radian = Degree × (π / 180)
Degree = Radian × (180 / π)
Gradian = Degree × (10 / 9)
What Is the Angle Converter (All-in-One)?
This angle converter takes a single value and instantly shows it in degrees, radians, and gradians side by side — the three units used across mathematics, navigation, surveying, and engineering to describe rotation and angular position.
Degrees trace back roughly 4,000 years to Babylonian astronomers, who worked in a base-60 (sexagesimal) counting system and divided a full circle into 360 parts — a number that divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, and 12, which made it practical for the era's hand calculations. Radians, by contrast, are a purely geometric unit: one radian is the angle formed when the arc length along a circle equals its radius, tying the unit directly to the shape rather than an arbitrary division. The word "radian" itself only entered print in 1873, coined by physicist James Thomson, although mathematicians had used the underlying idea for at least a century before that. Gradians (also called gons) are the newest of the three — French scientists introduced them during the 1790s metric reforms that also produced the meter and kilogram, defining a right angle as exactly 100 gradians so a full circle became a clean 400.
Because none of these systems share a common base, converting between them by hand means juggling π and awkward fractions. This calculator does that math instantly. If your work also involves distances or speeds — common in surveying and navigation — the Length Converter and Speed Converter cover those conversions too.
Angle Converter (All-in-One) Formula
- 180° = π radians
- 90° = π/2 radians
- 1 radian ≈ 57.2958°
How Is the Angle Converter (All-in-One) Calculated?
A full circle spans 360 degrees, 2π radians, or 400 gradians — three different ways of dividing the same rotation. Converting between them is a matter of scaling by the ratio between these full-circle values, using radians (the natural mathematical unit tied directly to a circle's radius) as the common base.
Radians work out to the "natural" unit for rotation because a circle's circumference is always 2π times its radius, so a full turn equals exactly 2π radians without introducing any arbitrary scaling constant — this is why calculus, physics, and most programming math libraries default to radians for angle calculations. Gradians instead follow decimal logic: because a right angle is fixed at exactly 100 gradians, any angle expressed in gradians can be read directly as a percentage of a right angle, which is what appealed to the metric-system reformers pushing for base-10 consistency across every unit of measurement.
Angle Converter (All-in-One) Example
An angle of 60° converts to 60 × (π/180) ≈ 1.047 radians, or 60 × (10/9) ≈ 66.67 gradians.
An angle of 3 radians converts to 3 × (180/π) ≈ 171.89°, or 3 × (200/π) ≈ 190.99 gradians.
A bearing of 350 gradians — the kind of value you might see in an old European survey record — converts to 350 × (9/10) = 315°, or 350 × (π/200) ≈ 5.498 radians.
How to Use the Angle Converter (All-in-One)
Step 1
Enter an angle value.
Step 2
Select the unit it's currently measured in (degree, radian, or gradian).
Step 3
View the converted angle across all three supported units instantly.
Step 4
Switch the input unit to see how the same rotation reads in a different system.
Step 5
Use the radian figure directly in code or a scientific calculator's trig functions, which almost always expect radians rather than degrees.
Step 6
Check the gradian output when working with older European surveying or cadastral records.
Benefits
- Converts degrees, radians, and gradians in one pass instead of running three separate lookups.
- Uses exact π-based math, removing rounding errors that creep into manual calculation.
- Helps you sanity-check trigonometric function inputs before plugging them into code.
- Covers gradians, a unit most everyday converters leave out entirely.
- Turns your conversion into a clean, ready-to-post image card instead of a screenshot of raw numbers.
- Runs entirely in your browser, with no sign-up or app download required.
Common Angle Converter (All-in-One) Scenarios
Scenario 1
Converting a textbook or exam answer from radians back to degrees to double-check it.
Scenario 2
Preparing angle inputs for trigonometric functions in Python, JavaScript, or a spreadsheet formula.
Scenario 3
Reading historical European survey or cadastral records that record bearings in gradians.
Scenario 4
Converting rotation values between unit conventions in CAD or 3D modeling software.
Scenario 5
Cross-checking a compass bearing given in degrees against a gradian-based instrument reading.
Scenario 6
Working through physics, robotics, or engineering coursework that mixes angular units.
Understanding Your Result
All three values represent the exact same angle, just expressed on different scales — 360°, 2π radians, and 400 gradians all correspond to one full rotation, so none of the three numbers is more "correct" than the others.
A few reference points are worth memorizing, since they come up constantly in geometry and trigonometry:
| Degrees | Radians | Gradians |
|---|---|---|
| 30° | π/6 ≈ 0.5236 | 33.33 |
| 45° | π/4 ≈ 0.7854 | 50 |
| 90° | π/2 ≈ 1.5708 | 100 |
| 180° | π ≈ 3.1416 | 200 |
| 360° | 2π ≈ 6.2832 | 400 |
Tips
- Remember that most programming languages and scientific calculators expect radians for trig functions, not degrees.
- Use exact fractions of π (like π/2, π/4) when precision matters more than decimal approximations.
- Gradians are rarely used outside surveying and certain European contexts — degrees or radians cover most everyday needs.
- When comparing rotation values across two software tools, check each tool's documentation for its assumed unit before trusting a shared numeric value.
- For angles beyond 360°, remember they still represent a valid rotation (more than one full turn) rather than an error.
Common Mistakes
- Feeding a degree value into a function that expects radians (a common bug in trigonometric code).
- Confusing gradians (400 per circle) with degrees (360 per circle) when reading old surveying data.
- Rounding intermediate radian values too early, compounding error in further calculations.
- Assuming gradians and grads used in artillery/military "mils" are the same unit — they are different systems (mils typically divide a circle into 6,400 parts).
- Forgetting that negative angles are valid and represent rotation in the opposite direction, not an invalid input.
Frequently Asked Questions
What is a radian?
A radian is the standard unit of angle measurement in mathematics, defined as the angle subtended when the arc length equals the radius of a circle. It is the default angle unit in calculus, physics, and most programming languages.
Why are radians used instead of degrees in math?
Radians simplify calculus and trigonometric formulas, since they relate directly to the radius and arc length of a circle without an arbitrary scaling factor like 360 — derivatives of sine and cosine only take their simplest form when the angle is in radians.
What is a gradian?
A gradian (or gon) divides a full circle into 400 units instead of 360, making a right angle exactly 100 gradians. It was introduced during the French metric reforms of the 1790s and is occasionally used in surveying and some European countries today.
How many radians are in a full circle?
A full circle is exactly 2π radians, approximately 6.2832 radians — this follows directly from the fact that a circle's circumference is 2π times its radius.
Do programming languages use degrees or radians?
Most programming languages (JavaScript, Python, Java, C, and others) use radians for trigonometric functions like sin() and cos(), so degree values usually need converting first with a formula like radians = degrees × (π/180).
What is a right angle in each unit?
A right angle is 90 degrees, π/2 radians (≈1.5708), or 100 gradians — the gradian value is exact by definition, while the degree and radian values are exact and approximate respectively.
Can angle values be negative?
Yes, negative angles represent rotation in the opposite (clockwise) direction and convert the same way as positive angles, simply preserving the sign across degrees, radians, and gradians.
Which unit is most common in everyday use versus math and engineering?
Degrees are most common in everyday contexts like navigation, geography, and construction, while radians are the standard unit in higher mathematics, physics, and most programming math libraries.
How do I convert a negative angle?
This converter handles negative angles the same way as positive ones, simply preserving the sign across all three units — a negative angle typically represents a clockwise or reverse rotation.
Can I share this conversion as an image?
Yes — tap Share and, on supported devices, your conversion is shared as a branded image card, not just a text link.
Why does a circle have 360 degrees instead of a rounder number like 100 or 1,000?
The 360-degree convention comes from ancient Babylonian astronomy, which used a base-60 number system; 360 divides evenly by many small numbers (2 through 12 except 7 and 11), which made angle calculations by hand much easier than a round base-10 number would have.
Where are gradians still actually used today?
Gradians persist mainly in French and some other European land-surveying and cadastral systems, in certain civil engineering software, and in a handful of scientific calculators that offer degree/radian/gradian mode alongside the more common two.
Is there a simple way to estimate radians from degrees without a calculator?
A rough shortcut is dividing the degree value by 57, since one radian is close to 57.3 degrees — it won't be exact, but it's a fast sanity check before trusting a precise conversion.
How precise is this converter's rounding?
Results are shown to two decimal places for readability; the underlying calculation uses full floating-point precision internally, so you can trust the displayed value for typical classroom, engineering, or hobby use.
References
Important Information
Conversions are exact mathematical relationships based on π.
Last updated: July 25, 2026