About this tool
What Is an Angle Converter?
An angle converter transforms angular measurements from one unit system to another — such as converting degrees to radians or gradians to arcseconds. Different fields of science, engineering, and mathematics use different angular units, and converting between them requires specific mathematical formulas.
A full circle can be expressed as 360 degrees, 2π radians, 400 gradians, 21,600 arcminutes, 1,296,000 arcseconds, approximately 6,283.19 milliradians, or exactly 1 turn. This converter handles all interconversions between these seven standard angular measurement systems.
Understanding Angular Measurement Units
Degrees (°): The most commonly used angular unit. A full circle is 360°, a right angle is 90°, and a straight line is 180°. Degrees are used in everyday geometry, navigation, and most practical applications.
Radians (rad): The SI (International System) unit for angles used in mathematics and physics. A full circle equals 2π radians (approximately 6.2832 rad). Radians are essential in calculus because they simplify the derivatives and integrals of trigonometric functions — the derivative of sin(x) is cos(x) only when x is in radians.
Gradians (gon): Also called "gons" or "grades." A full circle is 400 gradians, making a right angle exactly 100 gradians. This decimal-friendly system was developed in France and is still used in European land surveying and some engineering applications.
Arcminutes (′) and Arcseconds (″): Subdivisions of degrees based on the sexagesimal (base-60) system. One degree equals 60 arcminutes, and one arcminute equals 60 arcseconds. Arcseconds are primarily used in astronomy to measure very small angles — the apparent size of stars, planetary positions, and telescope precision.
Milliradians (mrad): One-thousandth of a radian. Used in military applications for ballistic calculations, rifle scope adjustments, and surveying. One milliradian at 1,000 meters corresponds to approximately 1 meter of displacement.
Turns: One turn equals one complete revolution (360° or 2π radians). Used in some engineering and physics contexts where counting complete rotations is more intuitive than measuring in degrees or radians.
Key Conversion Formulas
| Conversion | Formula | Example |
|---|---|---|
| Degrees → Radians | degrees × (π // 180) | 180° = π rad ≈ 3.1416 rad |
| Radians → Degrees | radians × (180 // π) | 1 rad ≈ 57.2958° |
| Degrees → Gradians | degrees × (400 / 360) | 90° = 100 gon |
| Degrees → Arcminutes | degrees × 60 | 1° = 60′ |
| Degrees → Arcseconds | degrees × 3,600 | 1° = 3,600″ |
| Degrees → Milliradians | degrees × (π // 180) × 1000 | 1° ≈ 17.4533 mrad |
| Degrees → Turns | degrees / 360 | 360° = 1 turn |
Why Radians Matter in Mathematics
The radian is the natural unit for measuring angles because it creates the simplest mathematical relationships. In calculus:
- d/dx sin(x) = cos(x) (only in radians)
- d/dx cos(x) = -sin(x) (only in radians)
- The limit of sin(x)/x as x→0 equals 1 (only in radians)
If angles were measured in degrees, these formulas would require an additional conversion factor of π/180, making every trigonometric calculation more complex. This is why virtually all programming languages (JavaScript, Python, C++) use radians as the default unit for trigonometric functions like Math.sin(), Math.cos(), and Math.atan2().
Common Angle Conversion Mistakes
Mistake 1: Confusing degrees and radians in calculators. Scientific calculators and programming languages default to radians. If you calculate sin(90) expecting 1, you get sin(90 radians) ≈ 0.894. You need sin(π/2) for the expected result.
Mistake 2: Forgetting arcminute/arcsecond conversion factors. There are 60 arcminutes in a degree and 60 arcseconds in an arcminute, giving 3,600 arcseconds per degree — not 100 or 360.
Mistake 3: Mixing up milliradians and degrees. One milliradian is approximately 0.0573°, not 0.001°. The relationship goes through radians: 1 mrad = 0.001 rad = 0.001 × (180/π) degrees.
Mistake 4: Assuming gradians equal degrees. A 90° right angle is 100 gradians, not 90. The 10% difference accumulates in calculations.
Practical Usage Examples
Degrees to Radians
Convert 45 degrees to radians.
45° × (π/180) = 0.785398 radiansRadians to Degrees
Convert π/6 radians to degrees.
0.523599 rad × (180/π) = 30.000000 degreesStep-by-Step Instructions
Step 1: Enter Your Angle Value. Type the numeric value you want to convert. The converter accepts decimals and negative values for angles in any direction.
Step 2: Select the Source Unit. Choose the unit your angle is currently measured in: degrees, radians, gradians, arcminutes, arcseconds, milliradians, or turns.
Step 3: Select the Target Unit. Choose the unit you want to convert to. All 7 units are available as conversion targets.
Step 4: Read the Result. The converted value is displayed with 6 decimal places of precision, suitable for scientific, engineering, and academic applications.
Step 5: Copy the Result. Click the copy button to use the converted value in your calculations, CAD software, spreadsheet, or academic work.
Core Benefits
6-Decimal Precision: Results are calculated to 6 decimal places, meeting the precision requirements of engineering, surveying, and scientific applications.
7 Unit Types Supported: Convert between degrees, radians, gradians, arcminutes, arcseconds, milliradians, and turns — covering mathematics, physics, surveying, astronomy, and military applications.
Instant Calculation: Results appear immediately without page reloads or waiting. The conversion uses JavaScript math functions that execute in microseconds.
Reference for Students: Verifies textbook calculations involving trigonometric conversions, radian-degree relationships, and angular measurement problems.
No Software Needed: Replaces the need for dedicated scientific calculators, MATLAB, or manual formula application for common angle conversions.
Frequently Asked Questions
Multiply the degree value by π/180 (approximately 0.01745). For example, 90° × (π/180) = π/2 radians ≈ 1.5708 radians. This formula works because a full circle of 360° equals 2π radians, so the ratio is π/180.
One radian equals approximately 57.2958 degrees. To convert radians to degrees, multiply by 180/π. A full circle is 2π radians, which equals 360 degrees.
A gradian (also called a gon or grade) is an angular unit where a full circle equals 400 gradians. A right angle is exactly 100 gradians. Gradians are used primarily in European land surveying and some civil engineering applications because the decimal-based system simplifies field calculations.
There are exactly 3,600 arcseconds in one degree. The conversion chain is: 1 degree = 60 arcminutes, and 1 arcminute = 60 arcseconds, so 60 × 60 = 3,600 arcseconds per degree.
Milliradians (mrad) are used in military ballistics, rifle scope adjustments, and precision surveying. The practical advantage: at a distance of 1,000 meters, 1 milliradian corresponds to approximately 1 meter of lateral displacement, making range and angle calculations intuitive for targeting.
Radians produce the simplest mathematical relationships in calculus and trigonometry. The derivative of sin(x) equals cos(x) only when x is in radians. Since programming language math libraries are built on these fundamental formulas, they use radians as the default unit for functions like sin(), cos(), and atan2().
An arcminute is 1/60th of a degree. An arcsecond is 1/60th of an arcminute (or 1/3600th of a degree). Arcseconds are extremely small angles used primarily in astronomy to measure star positions, telescope resolution, and satellite tracking precision.
Results are calculated to 6 decimal places using JavaScript double-precision floating-point arithmetic (IEEE 754), which provides approximately 15-16 significant digits internally. This exceeds the precision requirements of virtually all practical engineering and scientific applications.
One turn equals one complete revolution — 360 degrees, 2π radians, or 400 gradians. Turns are used in some physics and engineering contexts where counting full rotations is more intuitive, such as describing rotational speed (turns per minute instead of degrees per second).
Yes. Negative angles represent rotation in the opposite direction (typically clockwise vs. counterclockwise). The converter handles negative values correctly — for example, -90 degrees converts to -π/2 radians (-1.570796 rad).