Angle Converter

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degrees • radians • gradians • arcmin • arcsec

Angle Value ⓘ

From

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Enter an angle value to convert

How to Use the Angle Converter

Enter an angle value — e.g. 90 for a right angle, or 3.14159 for π radians.
Select From unit — degrees (everyday/navigation), radians (calculus/physics), gradians (surveying), arcmin/arcsec (astronomy).
Read the result — degrees, radians, gradians, and the exact π-form all show at once.
For trig functions — most calculators and programming languages use radians by default. Convert degrees → radians first.

Benefits

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Trigonometry & Calculus

Convert degrees to radians for sin, cos, tan in calculus, where radians are required for derivatives and series.

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Programming & CNC

JavaScript, Python, and C use radians in Math.sin(). CNC and CAD often use degrees. Convert to avoid errors.

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Astronomy & Optics

Telescopes and star positions use arcminutes and arcseconds — 1 degree = 60 arcmin = 3,600 arcsec.

Angle Conversion Guide — Degrees, Radians, Gradians

Angles can be measured in several units, and the most common conversion is between degrees and radians. A full circle is 360 degrees, 2π radians (approximately 6.2832), 400 gradians, or 1 turn. The key relationship: 1 radian = 180/π ≈ 57.2958 degrees, and 1 degree = π/180 ≈ 0.01745 radians. Degrees are intuitive for everyday use (a right angle is 90°), but radians are the natural unit in higher mathematics because they make calculus formulas clean — the derivative of sin(x) is cos(x) only when x is in radians.

Common angles convert to memorable radian values: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π. Gradians (also called gons) divide a right angle into 100 parts and are used in surveying and some European engineering. For precise work in astronomy and navigation, degrees are subdivided into arcminutes (1° = 60′) and arcseconds (1′ = 60″), so 1 degree = 3,600 arcseconds. Milliradians (mrad) are used in ballistics and optics for fine angular adjustments.

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Common Angle Conversions

30° = 0.5236 rad = π/6. 45° = 0.7854 rad = π/4. 60° = 1.0472 rad = π/3. 90° = 1.5708 rad = π/2. 180° = 3.1416 rad = π.

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Why Radians in Calculus

Radians are dimensionless (arc length ÷ radius), making trig derivatives and Taylor series work cleanly. d/dx sin(x) = cos(x) requires radians. This is why programming Math functions use radians.

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Surveying & Gradians

1 gradian = 1/400 of a circle = 0.9°. A right angle = 100 gon. Used in continental European surveying and some calculators (GRAD mode).

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Astronomy Precision

1° = 60 arcmin = 3,600 arcsec. The full Moon spans ~30 arcmin. Hubble resolves ~0.05 arcsec. Star coordinates use arcsec precision.

Angle Conversion FAQ

How do you convert degrees to radians? ▾

Multiply degrees by π/180 (≈ 0.0174533). So 90° × π/180 = π/2 ≈ 1.5708 radians. 180° = π radians. To go the other way, multiply radians by 180/π (≈ 57.2958). This converter does it instantly and also shows the exact π-fraction form.

How many radians in a full circle? ▾

A full circle = 2π radians ≈ 6.28319 radians = 360 degrees. A half circle = π radians = 180°. A quarter circle (right angle) = π/2 radians = 90°. Radians measure the arc length divided by the radius, so one full revolution traces an arc of 2πr around a circle of radius r.

What is a gradian (gon)? ▾

A gradian (also called a gon or grade) divides a right angle into 100 equal parts, so a full circle = 400 gradians and 1 gradian = 0.9 degrees. It's used in surveying and some European engineering because it simplifies percentage-grade slope calculations. Most scientific calculators have a GRAD mode alongside DEG and RAD.

Why do programming languages use radians? ▾

Functions like Math.sin(), Math.cos() in JavaScript, Python, Java, and C all expect angles in radians, not degrees. This is because radians are the mathematically natural unit for trigonometric functions and their derivatives. If you have an angle in degrees, convert first: radians = degrees × Math.PI / 180. Forgetting this is one of the most common bugs in graphics and game code.