📂Geometry

Radian

Definition

The angle of a sector with radius $r$ and arc length $\ell$ is called $\theta$ $\text{rad}$. Here $\text{rad}$ is read as radian.

Explanation

Since it is a value obtained by dividing a length by a length, it is a dimensionless quantity. Therefore the unit is usually omitted in practice. Angle values without units are, by default, in radians. The unit circle has radius $1$, so in this case the radian measure equals the arc length. Hence the circumference of the unit circle $2\pi$ can be seen to be equal to $360^{\circ}$. The relationship with degrees is as follows.

$$ 1 \text{rad} = \dfrac{180^{\circ}}{\pi} \approx = 57.2958^{\circ} $$

$$ 1^{\circ} = \dfrac{\pi}{180} \text{rad} \approx 0.0175 \text{rad} $$

In physics, the radian is also a typical example of a dimensionless quantity.