Angle Converter
Convert between degrees, radians, gradians, arcminutes, arcseconds, and revolutions.
Angle Converter
Enter an angle value and select its unit β all other units update instantly.
Quick References
How to Convert Between Degrees, Radians, Gradians, and Arcminutes
Your Python script calls math.sin(30) and returns 0.5, but the formula expects radians, not degrees. The correct call is math.sin(math.radians(30)). If you are unsure whether your calculator or code expects degrees or radians, type 30 into this converter, select degrees, and see that it equals 0.5236 radians. The copy button lets you paste the radian value straight into your code.
The converter handles 7 units: degrees, radians, gradians, arcminutes, arcseconds, revolutions, and NATO mils. A surveyor reading a European blueprint showing 100 grad needs to know that equals 90 degrees. An astronomer measuring a star separation of 120 arcseconds needs to express it as 0.0333 degrees or 0.000579 radians for a telescope pointing calculation. The quick-reference row shows common angles: a right angle at 90 degrees, a straight angle at 180 degrees, a full circle at 360 degrees.
Programming where the wrong angle unit produces wrong results
JavaScript, Python, and C all expect radians for Math.sin, Math.cos, and Math.tan. A developer computing sin(90) expecting 1.0 gets 0.8940 instead, because 90 radians is not 90 degrees. The correct value is sin(pi/2) = 1.0. This converter shows that 90 degrees = 1.5708 radians, so the correct code is Math.sin(1.5708) or Math.sin(Math.PI / 2). Getting this wrong produces silently incorrect results, not errors.
A land surveyor in France uses a theodolite that reads in gradians. The measured angle is 150 grad. The US construction drawings require degrees. Converting: 150 grad x 0.9 = 135 degrees. If the surveyor mistakenly treats 150 grad as 150 degrees, the building foundation would be laid out at the wrong angle, requiring expensive rework.
A forward observer reports a target 5 mils to the right. The gunner needs to adjust the aim. One mil equals 0.05625 degrees, so 5 mils = 0.28125 degrees. At 1 km distance, 5 mils corresponds to approximately 5 metres of lateral offset. The converter helps translate between the observer mil reading and the gunner degree adjustment.
How the Angle Converter Handles Degrees, Radians, and Gradians
Every unit in the converter is stored as a multiple of one degree. When you enter 1 radian, the tool multiplies by 180/pi to get 57.29578 degrees, then divides by each target unit's factor. The conversion to gradians gives 63.66198 grad. Because all conversions pass through degrees in one step, there is no rounding drift from chaining through intermediate units.
Why radians use an irrational conversion factor
The radian is defined as the angle subtended by an arc equal in length to the radius. This makes 1 full circle = 2pi radians = 360 degrees, so 1 radian = 180/pi degrees. Because pi is transcendental (irrational), the radian factor can never be expressed as an exact decimal. JavaScript's Math.PI provides 15 decimal digits of precision, which is sufficient for any practical calculation. The other units use exact rational factors: 1 grad = 0.9 degrees, 1 arcminute = 1/60 degree, 1 arcsecond = 1/3600 degree.
The NATO mil divides a circle into 6,400 equal parts (1 mil = 9/160 degree = 0.05625 degree). This number was chosen because 6,400 is close to 6,283 (2pi x 1,000), making mils approximately equal to milliradians. The approximation error is about 0.01%, which is acceptable for military gunnery where the mil allows quick range estimation: 1 mil corresponds to approximately 1 metre width at 1 km distance.
Why gradians exist and where they are still used
The gradian was introduced during the French Revolution as part of the metric system. It divides a right angle into exactly 100 parts, so a full circle = 400 grad. This makes percentage-based slope calculations trivial: a 50 grad slope rises 50 units for every 100 units horizontal, which is a 50% grade. Gradians are still used in some European surveying equipment (particularly French and Swiss theodolites) and in certain geodetic coordinate systems. Outside these niches, gradians are rarely encountered.
The converter formats results dynamically based on magnitude. Arcseconds for astronomical measurements display with 8-10 decimal places. Degrees and radians use 6-8 decimal places. Revolutions use 2-4 decimal places. Very large or very small values switch to scientific notation. This keeps the output readable whether you are converting arcsecond-scale telescope offsets or multi-revolution motor speeds.
Frequently Asked Questions
How many radians is 90 degrees?
90 degrees equals pi/2 radians, approximately 1.5708 rad. The conversion formula is radians = degrees x pi / 180. For quick mental estimates: 90 deg = 1.57 rad, 180 deg = 3.14 rad, 360 deg = 6.28 rad. These are the three most common conversions in physics and engineering.
Why do programming languages use radians instead of degrees?
Radians simplify calculus. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, an extra factor of pi/180 appears in every derivative and integral. This is why all major languages (JavaScript, Python, C, Java) use radians for their trigonometric functions. If your code uses degrees, you must convert first: radians = degrees x pi / 180.
What is a gradian used for?
Gradians (grad) divide a right angle into 100 parts, so a full circle = 400 grad. They are used in some European surveying equipment and geodetic calculations. A 50 grad slope equals exactly 50% grade. Outside surveying, gradians are rarely encountered. If you see an angle in gradians, multiply by 0.9 to get degrees.
How do I convert arcseconds to degrees?
Divide arcseconds by 3,600. One degree = 60 arcminutes = 3,600 arcseconds. So 7,200 arcseconds = 2 degrees. For quick estimates: 3,600 arcsec = 1 deg, 60 arcsec = 0.0167 deg. Arcseconds are used in astronomy for telescope pointing and stellar coordinate measurements.
What angle does a clock hand move per minute?
The minute hand moves 360 degrees in 60 minutes, which is 6 degrees per minute. In radians, that is 0.1047 rad/min. In gradians, that is 6.6667 grad/min. The hour hand moves 30 degrees per hour (360/12), which is 0.5 degrees per minute.