Angle Converter

An angle is the figure formed by two rays sharing a common endpoint (the vertex). It measures the amount of rotation between the rays. We classify angles by their size:

How to Convert Degrees to Radians

The formula: radians = degrees × (π / 180). The conversion factor π/180 ≈ 0.017453.

To go in reverse, multiply radians by 180/π. For example, π/3 radians × 180/π = 60°.

How to Convert DMS to Decimal Degrees

DMS stands for Degrees, Minutes, Seconds — a sexagesimal angle format used in surveying, navigation, and astronomy. Each degree subdivides into 60 minutes (each marked with '), and each minute subdivides into 60 seconds (marked with ").

To convert back: take the integer part as degrees, multiply the fractional part by 60 to get minutes, then take the fractional part of that and multiply by 60 to get seconds. The DMS field in the calculator above does this both directions instantly.

Why Radians Matter — The Arc Length Connection

Radians aren't arbitrary — they're defined so that the arc length on a circle equals the radius times the angle. The formula is wonderfully clean:

For an angle of 1 radian on a circle of radius 1, the arc length is exactly 1. That's the geometric definition. To explore this in detail, try our arc length calculator.

Other Angle Units: Gradians, Milliradians, and Mils

A few less-common angle units you may encounter:

• Gradian (gon): 1/400 of a circle. So 1 gon = 0.9° and a right angle = 100 gons. Used in some European surveying.
• Milliradian (mrad): 1/1000 of a radian. Used in optics, ballistics, and engineering. There are 2000π ≈ 6283 mrads in a full circle.
• NATO mil: A military approximation where the circle is divided into 6400 mils (instead of 6283), so the math comes out cleanly for mortar and artillery range estimation.
• Arcminute / Arcsecond: Subdivisions of a degree. 1° = 60', 1' = 60". Widely used in astronomy (e.g., the angular size of stars).

Understanding Angle Measurement

Angles can be measured in multiple units, with degrees and radians being the most common in mathematics and science. Understanding both systems and converting between them is essential for trigonometry, calculus, physics, and engineering applications.

Degrees

Degrees are the most intuitive angle measurement for most people. A full circle is divided into 360 degrees, possibly originating from ancient Babylonian astronomy which used a base-60 number system. Common angles include 90° (right angle), 180° (straight line), and 360° (full rotation). Degrees are ideal for everyday applications, navigation, and introductory mathematics. For more on understanding degrees and radians conversion, check out additional resources.

Radians

Radians measure angles based on the radius of a circle. One radian is the angle formed when the arc length equals the radius. A full circle contains 2π radians (approximately 6.28). Radians are the natural unit for calculus and physics because they simplify many mathematical formulas. For example, the derivative of sin(x) is cos(x) only when x is in radians. Learn more about converting between degrees and radians for practical applications.

Complete Angle Conversion Table (0°–360° in 15° increments)

Conversion Factor Quick Reference

When to Use Each Unit

Use Degrees for: Navigation, construction, everyday measurements, introductory geometry, and when communicating with non-technical audiences. Degrees are more intuitive for most people.

Use Radians for: Calculus, physics, advanced mathematics, computer programming, and any situation involving derivatives or integrals of trigonometric functions. Radians make mathematical formulas simpler and more elegant.