π Angle Converter
Degrees, radians, gradians, arcminutes, and arcseconds with live results.
All units
Common Angles Quick Reference
| Degrees | Radians | Gradians | Description |
|---|---|---|---|
| 0Β° | 0 | 0 | Zero angle |
| 30Β° | Ο/6 β 0.5236 | 33.33 | |
| 45Β° | Ο/4 β 0.7854 | 50 | Diagonal |
| 60Β° | Ο/3 β 1.0472 | 66.67 | |
| 90Β° | Ο/2 β 1.5708 | 100 | Right angle |
| 180Β° | Ο β 3.1416 | 200 | Straight angle |
| 270Β° | 3Ο/2 β 4.7124 | 300 | Reflex quarter |
| 360Β° | 2Ο β 6.2832 | 400 | Full turn |
How to Use This Angle Converter
Enter any angle value, pick the unit it is in, and every other unit updates instantly. Angles show up everywhere: geometry homework, trigonometry, navigation bearings, CNC machining, game development, and construction layout. The most common stumbling block is mixing up degrees and radians on a calculator, which this tool sidesteps by showing both side by side.
Type the angle value into the Value field. Decimals are fine, and negative angles work too.
Choose the starting unit from the dropdown: degrees, radians, gradians, arcminutes, arcseconds, or full turns.
Read every converted unit in the results panel. Use the Copy button to grab the main value for your homework or code.
Angle Units Explained
Degrees split a full circle into 360 parts and are the everyday unit for construction, navigation, and school geometry. Radians are the mathematician's unit: one radian is the angle where the arc length equals the radius, and a full circle is exactly 2Ο radians. Calculus, physics formulas, and programming libraries like JavaScript's Math.sin() all expect radians, which is why degree-to-radian conversion is the single most common angle task.
Gradians (also called gons) divide a right angle into 100 parts, so a full circle is 400 gradians. They are popular in French surveying and some European engineering because right angles land on round numbers. Arcminutes and arcseconds are tiny subdivisions of a degree: one degree holds 60 arcminutes, and one arcminute holds 60 arcseconds. Astronomers, opticians, and GPS engineers live in these units because a single arcsecond of latitude is about 31 meters on the ground.
The Key Conversion Formulas
Memorize the first two and you can derive the rest. To convert 180 degrees to radians: 180 Γ Ο / 180 = Ο β 3.14159. Going the other way, 1 radian Γ 180 / Ο β 57.2958 degrees. Gradians are just degrees scaled by 10/9, because 360 degrees equals 400 gradians. Arcminutes and arcseconds are sexagesimal subdivisions inherited from Babylonian astronomy over 3,000 years ago.
Worked Example
A student needs sin(45Β°) but their calculator is in radian mode. Convert 45 degrees: 45 Γ Ο / 180 = Ο/4 β 0.7854 radians. Entering 0.7854 into the radian-mode calculator gives sin β 0.7071, the correct answer. Another common case: a machinist sees a print calling for 2,700 arcseconds of taper. Divide by 3,600 to get 0.75 degrees, a much friendlier number for the machine readout.
Degrees vs Radians: When to Use Which
Use degrees for anything humans read: blueprints, compass bearings, oven dials, screen rotations in CSS. Use radians for anything computed: trig functions in code, angular velocity in physics (radians per second), and calculus, where the derivative of sin(x) is cos(x) only when x is in radians. Gradians shine in surveying and land measurement. Arcminutes and arcseconds belong to astronomy, optics, and precision navigation.
Frequently Asked Questions
How do I convert degrees to radians?
Multiply degrees by Ο/180 (about 0.0174533). So 90 degrees Γ 0.0174533 β 1.5708 radians. This converter does it instantly and shows gradians, arcminutes, and arcseconds at the same time.
Why do calculators have degree and radian modes?
Because trig functions need to know which unit the angle is in. sin(90) is 1 in degree mode but about 0.894 in radian mode. Forgetting to switch modes is the number one source of wrong answers in trig homework.
What is a gradian?
A gradian (gon) divides a right angle into 100 equal parts, making a full circle 400 gradians. It is used in surveying and some European engineering because perpendicular lines land on round numbers like 100 and 200.
How many arcseconds are in a degree?
3,600. One degree contains 60 arcminutes, and each arcminute contains 60 arcseconds. Astronomers use arcseconds because stars are so far away that even tiny angles matter.
What is the difference between a radian and a degree?
A degree is 1/360 of a circle, an arbitrary but convenient human unit. A radian is defined by the circle itself: it is the angle subtended when the arc length equals the radius, so a full circle is exactly 2Ο radians. Math and physics formulas assume radians.