Angle Unit Converter (degrees, radians, gradians)

Converts an angle into degrees, radians, gradians, arcminutes, arcseconds and turns. A straight angle of 180 degrees is π radians, one degree is 60 arcminutes, and one arcminute is 60 arcseconds.

Explanation

Enter an angle and choose its unit, and every other unit follows. Degrees convert to radians exactly as easily as radians convert back.

The definitions

180^\circ = \pi\ \text{rad}

Degrees to radians: multiply by $\dfrac{\pi}{180}$
Radians to degrees: multiply by $\dfrac{180}{\pi}$ , so one radian is about 57.2958°
1° = 60 arcminutes (′) = 3600 arcseconds (″)
1 gradian = 0.9°, making a right angle exactly 100 gradians

Example

Convert 90 degrees.

90 \times \dfrac{\pi}{180} = \dfrac{\pi}{2} = 1.5708\ \text{rad}

That is 100 gradians, or a quarter of a turn.

What a radian really is

One radian is the angle that cuts off an arc as long as the radius. Since the circumference is $2\pi r$ , a full turn is $2\pi$ radians and a half turn is $\pi$ .

Degrees rest on a human decision to slice the circle into 360 parts. A radian is a ratio of two lengths, so it carries no arbitrary choice at all — which is why calculus insists on it. The derivative of $\sin x$ is $\cos x$ only when $x$ is in radians.

The classic bug

Every programming language's `Math.sin()` expects radians. Feeding it degrees gives nonsense.

Wrong: `Math.sin(30)` → −0.988, the sine of 30 radians
Right: `Math.sin(30 * Math.PI / 180)` → 0.5