📂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 length divided by a length, it is a dimensionless quantity. Therefore the unit is usually omitted. An angle value with the unit omitted is, by convention, expressed in radians.

$$ \theta = \dfrac{\ell}{r} = \dfrac{\text{arc length}}{\text{radius}} \implies \dim \theta = \dfrac{\mathsf{L}}{\mathsf{L}} = 1 $$

For the unit circle the radius is $1$, so in this case the numerical value in radians equals the arc length. Hence the circumference of the unit circle, $2\pi$, is equal to $360^{\circ}$. The relationship with the other unit of angle, the degree, is as follows.

$$ 1 \text{ rad} = \left( \dfrac{180}{\pi} \right)^{\circ} \approx 57.2958^{\circ} $$

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