Angle Converter

Maths classes use degrees, programming libraries use radians, surveyors use gradians and astronomers use arcminutes and arcseconds. This converter moves an angle between degrees, radians, gradians, turns, arcminutes, arcseconds and milliradians, and shows every unit at once.

How to use it

1. Enter an angle.
2. Choose the unit you have and the one you want.
3. Read the headline result and the table.
4. Use Swap units to go the other way.

When it helps

• Converting degrees to radians before feeding them to a sine or cosine function.
• Reading surveying or astronomy values.
• Working with rotation values in graphics and game code.
• Checking trigonometry homework.

Good to know

• 180 degrees is pi radians, about 3.14159, and a full turn is 2 pi radians, 360 degrees or 400 gradians.
• A degree has 60 arcminutes and an arcminute has 60 arcseconds.
• Most programming languages expect radians, which is the usual source of "my angle is wrong" bugs.

How the conversion works

A full circle is 360 degrees, 2 pi radians, 400 gradians or one turn. Everything converts through degrees: a radian is 180 divided by pi, about 57.2958 degrees, a gradian is 0.9 degrees, an arcminute a sixtieth of a degree, and an arcsecond a sixtieth of that.

Radians are the natural unit in mathematics, because an angle in radians is the length of the arc it cuts from a circle of radius one. That's why formulas for derivatives of sine and cosine are tidy in radians and messy in degrees.

Worked example: a programming bug

You call a sine function with 90 and expect 1, but you get 0.894. The function expects radians, and 90 radians is a very different angle. Convert first: 90 degrees is 1.5708 radians, and the sine of that is 1.

In the other direction, a game character rotating at 2 radians per second turns about 114.6 degrees per second, or a full circle in about 3.14 seconds, which is exactly pi.

Navigation and astronomy

Sky positions use degrees, minutes and seconds. The Moon is about half a degree across, or 30 arcminutes, and a degree of latitude on Earth is about 111 kilometres, so an arcminute is roughly a nautical mile. That relationship is why nautical miles exist.

Surveyors sometimes use gradians because a right angle is exactly 100, making slopes easy to work out. Military mapping uses milliradians, where one milliradian covers about one metre at a distance of one kilometre.

Common angles to remember

30 degrees is pi over six, 45 is pi over four, 60 is pi over three, 90 is pi over two, 180 is pi, and 360 is two pi. Memorising these saves a lot of time in trigonometry.

If a converted result looks like a long decimal close to one of these, you've probably got a standard angle, and you can express it as a fraction of pi.

Questions