How do you convert degrees to radians?

Multiply the degree value by π/180 (approximately 0.0174533). For example, 90 degrees × (π/180) = π/2 ≈ 1.5708 radians. This works because a full circle is both 360 degrees and 2π radians, so π/180 is the conversion ratio between the two systems.

Why do programming languages use radians instead of degrees?

Radians are the natural unit for calculus and trigonometry because they directly relate an angle to arc length on a unit circle (arc length = radius × angle in radians). This makes derivatives and integrals of trigonometric functions clean, without extra conversion constants, which is why virtually every programming language's math library standardizes on radians.

What is a gradian and why is it used?

A gradian (or gon) divides a full circle into 400 equal parts instead of 360, making a right angle exactly 100 gradians. This decimal-friendly system is used in some European surveying, civil engineering, and geodesy contexts because it simplifies certain field calculations compared to the 90-degree right angle of the degree system.

How many radians are in a full circle?

A full circle equals 2π radians, which is approximately 6.28319 radians. This corresponds to 360 degrees or 400 gradians — all three describe the same complete rotation, just measured in different unit systems.

What is 1 radian in degrees?

1 radian equals approximately 57.2958 degrees, calculated as 180/π. This is why a radian feels like an unusual, non-round number in degrees — it's defined geometrically (as the angle subtending an arc equal to the circle's radius) rather than by dividing the circle into a round number of parts.

How do you convert gradians to radians?

Multiply gradians by π/200 (approximately 0.0157080), since a full circle is 400 gradians and 2π radians, making the ratio 2π/400 = π/200. For example, 100 gradians (a right angle) × π/200 = π/2 ≈ 1.5708 radians.

Angle Converter

Unit Converters

Convert between degrees, radians, and gradians

Value

From

Degrees feel intuitive — a right angle is 90, a full circle is 360 — but most math and programming functions (sin, cos, tan in nearly every programming language) expect radians, not degrees. Mix them up and a calculation that should return 0.5 instead returns a wildly wrong number, a bug that trips up students and engineers alike. Add gradians, a less common but still-used unit in surveying and some European engineering contexts, and manual conversion becomes an easy place to make mistakes.

Our angle converter instantly converts between degrees, radians, and gradians in either direction. Enter a value in any of the three units and get accurate results using exact conversion formulas, not rounded approximations. Whether you're writing code that calls a trigonometric function, working through a physics or engineering problem, plotting a rotation in CAD software, or double-checking a surveying calculation, this tool removes the guesswork and gives you the precise converted value instantly.

Why Angle Converter Matters

Angle units aren't just a math-class abstraction — they show up constantly in real technical work. Programming languages including Python, JavaScript, C++, and Java implement trigonometric functions using radians by default, so any code working with angles measured in degrees (like a game's rotation value or a robotics joint angle) must convert to radians before calling sin(), cos(), or tan(), or the results will be silently wrong.

Engineering and physics rely on radians because they simplify calculus involving circular and rotational motion — angular velocity, arc length, and wave functions are all naturally expressed in radians. Navigation, surveying, and some European civil engineering traditions use gradians (also called gons), where a right angle is a clean 100 gradians instead of 90 degrees, making certain field calculations easier. Getting these conversions right matters in CAD modeling, robotics programming, land surveying, astronomy, and any 3D graphics or animation work where a wrong angle unit produces a visibly incorrect rotation.

The Angle Converter Formula, Explained

radians = degrees × (π/180); gradians = degrees × (10/9); degrees = radians × (180/π) = gradians × (9/10)

A full circle is 360 degrees, 2π radians (approximately 6.28319), or 400 gradians. Because degrees and radians both describe the same 360-degree circle, the conversion factor between them is π/180 ≈ 0.0174533: multiply degrees by π/180 to get radians, or multiply radians by 180/π ≈ 57.2958 to get degrees.

Gradians divide the circle into 400 equal parts instead of 360, which makes a right angle exactly 100 gradians — a convenient property for surveying and some European civil engineering work. To convert degrees to gradians, multiply by 10/9 (since 400/360 = 10/9). To convert gradians back to degrees, multiply by 9/10. To convert directly between radians and gradians, first convert to degrees as an intermediate step, or use radians × (200/π) = gradians.

Key reference points: 90° = π/2 rad ≈ 1.5708 rad = 100 grad. 180° = π rad ≈ 3.14159 rad = 200 grad. 270° = 3π/2 rad ≈ 4.71239 rad = 300 grad. 360° = 2π rad ≈ 6.28319 rad = 400 grad.

How to Use the Angle Converter: Step by Step

Enter the angle value
Type the numeric angle you want to convert, such as 45 or 1.5708.
Select the unit you're converting from
Choose degrees, radians, or gradians as your starting unit.
Select the target unit
Choose the unit you want to convert to — the tool applies the exact π-based formula automatically.
Read the converted result
The converted angle displays instantly with precise decimal accuracy, ready to use in code, drawings, or calculations.

Angle Converter Examples: Real-World Scenarios

Converting a Rotation Angle for Code

A game developer needs to rotate a sprite by 45 degrees, but the graphics library's rotate() function only accepts radians.

Angle:45 degrees

Formula:radians = degrees × (π/180)

Calculation

45 × (π/180) = 45 × 0.0174533 = 0.785398 radians

Result

The developer passes 0.785398 (approximately π/4) into the rotate() function to correctly rotate the sprite by 45 degrees.

Reading a Robotics Joint Angle

A robotic arm's controller reports a joint position as 1 radian and an engineer needs the equivalent in degrees to compare against the mechanical spec sheet.

Angle:1 radian

Formula:degrees = radians × (180/π)

Calculation

1 × (180/π) = 57.2958 degrees

Result

The joint is positioned at approximately 57.30 degrees, which the engineer confirms is within the arm's 0–180 degree operating range.

Converting a Survey Angle from Gradians

A land surveyor using European equipment reads a bearing angle of 150 gradians and needs it in degrees for a report using U.S. conventions.

Angle:150 gradians

Formula:degrees = gradians × (9/10)

Calculation

150 × 0.9 = 135 degrees

Result

The 150-gradian bearing converts to exactly 135 degrees, which the surveyor records in the final report.

Common Mistakes to Avoid

Feeding degree values directly into programming functions like sin() or cos() without converting to radians first — nearly all languages expect radians, so sin(90) in code returns a value close to 0.894 instead of the expected 1.
Confusing gradians with grads used in artillery or military mils — gradians (400 per circle) are distinct from angular mils (6,400 per circle), a different unit used in some military and tactical contexts.
Rounding π too early in multi-step calculations — using 3.14 instead of a fuller value of π introduces small errors that compound in trigonometric or navigation calculations requiring precision.

Tips & Tricks

Memorize the key benchmark: 180 degrees equals exactly π radians. From there, 90° = π/2, 60° = π/3, 45° = π/4, and 30° = π/6 — these fractions cover most common angles used in trigonometry.
When working in code, check your language's math library documentation — nearly all standard libraries (Python's math module, JavaScript's Math object, etc.) use radians exclusively for trig functions, with degrees requiring manual conversion.

Angle units are simple in concept but easy to mix up across math, code, and fieldwork. Use this converter to move confidently between degrees, radians, and gradians whenever you're programming, studying trigonometry, or working through an engineering or surveying calculation. For related geometry and measurement conversions, see our area and length converters.

Angle Converter — Frequently Asked Questions

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Authoritative References

NIST — The International System of Units (SI), Plane Angle

nist.gov — Official reference on the radian as the SI unit of plane angle

Wolfram MathWorld — Radian

mathworld.wolfram.com — Mathematical reference explaining the definition and derivation of radians

External links open in a new tab. OmniCalc.us is not affiliated with these organisations.

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