Convert Degrees to Radians (Step‑by‑Step with LaTeX)

Convert degrees to radians – step by step calculator

fundamental relation

\[ \Large{ \text{radians} = \text{degrees} \times \frac{\pi}{180}} \] Because \(360^\circ = 2\pi \text{ rad}\) ⇒ \(1^\circ = \frac{\pi}{180} \text{ rad}\)

one full rotation = \(360^\circ = 2\pi \text{ rad}\)

Examples

\[ \begin{aligned} 120^\circ &= 120 \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3} \approx 2.094 \text{ rad}, \\ -20^\circ &= -20 \times \frac{\pi}{180} = -\frac{20\pi}{180} = -\frac{\pi}{9} \approx -0.349 \text{ rad}, \\ 34^\circ 30' 45'' &= 34 + \frac{30}{60} + \frac{45}{3600} = 34.5125^\circ \rightarrow 34.5125 \times \frac{\pi}{180} \approx 0.692 \text{ rad}. \end{aligned} \]

Decimal degrees → radians

Enter angle as a decimal number (example: 30.0, –45.8)

Angle in degrees (decimal)

degrees

radians (decimal) 0.5236

as fraction of \(\pi\) ¹⁄₆\(\pi\) π

decimal places

↻ change decimal places and press Calculate again

step-by-step (decimal)

\( \text{radians} = \theta \times \frac{\pi}{180} \)

\( = 30 \times \frac{\pi}{180} \)

\( = \frac{30\pi}{180} \)

\( = \frac{1}{6}\pi \approx 0.5236 \text{ rad} \)

Degrees, minutes, seconds → radians

Example: \(10^\circ 12' 34''\) (positive or negative)

Angle: degrees, minutes, seconds

Degrees (°)

Minutes (′)

Seconds (″)

radians (decimal) 0.1994

decimal places

↻ decimal places applied after calculation

step-by-step (DMS)

\( \theta = 10^\circ 12' 34'' = 10 + \frac{12}{60} + \frac{34}{3600} = 10.20944^\circ \)

\( \text{radians} = 10.20944 \times \frac{\pi}{180} \)

\( = \frac{10.20944\pi}{180} \approx 0.056719\pi \)

\( \approx 0.1994 \text{ rad} \)

Why radians matter – applications

• arc length formula: \[ \text{arc} = \theta \cdot r \;\; , \;\; \theta\ \text{in radians} \].

• calculus: \[\frac{d}{dx}\sin x = \cos x \;\; , \;\; x \text{ in radians} \] In degrees you get a factor \(\frac{\pi}{180}\).

• series expansion: \[\sin x = x - \frac{x^3}{3!} + \cdots \;\; , \;\; x \text{ in radians} \].

More references and tools

angles from degrees to radians – interactive calculator with LaTeX steps, purple material theme, designed for learning.