Convert Angle Units
Angle Converter
Angle Conversion Chart
Key relationships between angle measurement systems. The SI unit is the radian (rad).
Degrees, Minutes & Seconds
| 1 full circle | = | 360° |
| 1° | = | 60 arcminutes (′) |
| 1 arcminute (′) | = | 60 arcseconds (″) |
| 1° | = | 3,600 arcseconds (″) |
Degrees ↔ Radians
| 1 full circle | = | 2π rad ≈ 6.28318 rad | exact |
| 1 rad | = | 180°/π ≈ 57.2958° | exact |
| 1° | = | π/180 ≈ 0.017453 rad | exact |
| 90° | = | π/2 rad | exact |
Other Systems
| 1 full circle | = | 400 gradians (gon) | exact |
| 1° | = | 10/9 gradians | exact |
| 1 full circle | = | 6,400 mils (NATO) | exact |
| 1 turn | = | 1 revolution = 360° | exact |
The relationship between degrees and radians is exact by mathematical definition: 180° = π radians. The gradian (also called gon or grade) divides the right angle into exactly 100 parts, making it useful for surveying. The NATO mil divides the circle into 6,400 parts.
Common Angles Reference
Frequently used angles and their values in different unit systems.
| Angle | Degrees (°) | Radians (rad) | Gradians (gon) | Turns | Common Use |
|---|---|---|---|---|---|
| 30° | 30 | π/6 ≈ 0.5236 | 33.333 | 1/12 | Trigonometry, clock (1 hour) |
| 45° | 45 | π/4 ≈ 0.7854 | 50 | 1/8 | Diagonal, isosceles right triangle |
| 60° | 60 | π/3 ≈ 1.0472 | 66.667 | 1/6 | Equilateral triangle interior |
| 90° | 90 | π/2 ≈ 1.5708 | 100 | 1/4 | Right angle, perpendicular |
| 120° | 120 | 2π/3 ≈ 2.0944 | 133.333 | 1/3 | Regular hexagon interior |
| 180° | 180 | π ≈ 3.1416 | 200 | 1/2 | Straight angle, half turn |
| 270° | 270 | 3π/2 ≈ 4.7124 | 300 | 3/4 | Three-quarter turn |
| 360° | 360 | 2π ≈ 6.2832 | 400 | 1 | Full rotation |
In trigonometry, angles of 30°, 45°, and 60° are called "special angles" because their sine, cosine, and tangent values can be expressed as simple fractions and square roots.
About Angle Units
Mathematics & Trigonometry
Degrees (°) are the most familiar unit for everyday angles. Radians (rad) are the standard in calculus, physics, and programming — most math libraries expect angles in radians. Key relation: π radians = 180°.
Surveying & Engineering
Gradians (gon) divide the right angle into exactly 100 parts, making slope calculations simpler. Used primarily in surveying in continental Europe. The NATO mil divides the circle into 6,400 parts for artillery and navigation.
Navigation & Astronomy
Arcminutes (′) and arcseconds (″) measure very small angles. Celestial coordinates use degrees, arcminutes, and arcseconds (DMS format). 1 nautical mile = 1 arcminute of latitude. The DMS ↔ Decimal converter handles this format.
Popular Angle Conversions
Degrees ↔ Radians
Degrees ↔ Other Units
Radians & Other Units
Need DMS format? Use our DMS ↔ Decimal Degrees converter.
Conversion Charts & Tables
Printable charts and reference tables for angle conversions:
Frequently Asked Questions
What is the difference between degrees and radians?
Degrees divide a full circle into 360 equal parts, while radians measure angles based on the radius of a circle. One full circle equals 360 degrees or 2π radians (approximately 6.2832 radians). To convert degrees to radians, multiply by π/180. Radians are the standard unit in mathematics and physics, while degrees are more common in everyday use, navigation, and construction.
Why are there 360 degrees in a circle?
The 360-degree system originated with the ancient Babylonians, who used a base-60 (sexagesimal) number system. The number 360 is highly divisible — it has 24 factors — making it practical for dividing circles into equal parts. It also approximates the number of days in a year, which was significant for early astronomical observations and calendar systems.
When should I use gradians instead of degrees?
Gradians (also called gons or grads) divide a right angle into 100 equal parts, making a full circle 400 gradians. They are primarily used in surveying, geodesy, and some European engineering standards because the decimal subdivision simplifies calculations. For most everyday and scientific applications, degrees or radians are preferred.
How do I measure angles without a protractor?
You can estimate common angles using your hand: a spread hand (thumb to pinky) is roughly 20 degrees, and a fist at arm's length covers about 10 degrees of sky. For precision, digital angle-measuring devices, smartphone apps with gyroscope sensors, or a simple plumb line with a printed scale can be used. In construction, speed squares and combination squares provide accurate angle references.
Related Converter Categories
- Frequency Converter — Angular frequency: ω = 2πf
- Length Converter — Arc length = radius × angle (in radians)
