Radians of a Circle

A radian is defined by the circle itself: it is the angle subtended at the centre by an arc whose length equals the radius.

$\theta = \frac{s}{r}$

Because a full circumference is $s = 2\pi r$, one complete turn measures $\dfrac{2\pi r}{r} = 2\pi$ radians. That $r$ cancels — which is why a radian is dimensionless, a pure ratio rather than a unit like a metre.

$2\pi \text{ rad} = 360°, \qquad \pi \text{ rad} = 180°, \qquad 1 \text{ rad} = \frac{180°}{\pi} \approx 57.2958°$

Why bother? Because the ratio definition is what makes calculus work. $\lim_{x\to 0}\frac{\sin x}{x} = 1$ and $\frac{d}{dx}\sin x = \cos x$ are true only in radians; in degrees the derivative picks up a stray factor of $\pi/180$. Every series expansion, every physics formula for angular velocity, assumes radians.

Multiply by the conversion factor that cancels the unit you are leaving:

$\text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}$

Arc length and sector area need $\theta$ in radians — the simple forms are false in degrees:

$s = r\theta, \qquad A_{\text{sector}} = \tfrac{1}{2}r^2\theta$

• Calculator left in the wrong mode. $\sin(30)$ is $0.5$ in degree mode and $-0.988$ in radian mode. Check DEG/RAD before every trig problem.
• Flipping the conversion factor. Going to radians multiplies by $\pi/180$ — the answer should carry a $\pi$. Going to degrees multiplies by $180/\pi$ and the $\pi$ should vanish.
• Using $s = r\theta$ with degrees. Convert first, or use $s = \frac{\pi r \theta_{\deg}}{180}$.
• Rounding $\pi$ too early. Keep exact multiples like $3\pi/4$ until the final line; $2.36$ hides the structure and loses precision.
• Thinking radians are "bigger" angles. One radian is about $57.3°$, so $2$ radians is roughly $115°$, not two full turns.
• Assuming $\pi$ always means $180°$ in an expression. In $\sin(\pi x)$, $\pi$ is a coefficient, not an angle.