Degrees to Radians Calculator

Degrees and radians are two ways of measuring the same thing — how far around a circle an angle sweeps. Degrees divide a full rotation into 360 equal parts, a convention inherited from Babylonian astronomy. Radians instead measure the angle by the length of the arc it subtends on a circle of radius 1: a full rotation sweeps an arc of 2 pi (the circumference of the unit circle), so 360 degrees equals 2 pi radians. That single relationship, radians = degrees x pi / 180, is exact and underpins every calculation on this page.

This calculator goes far beyond a simple conversion. It shows the exact pi fraction for all common angles, plots the angle on an interactive unit-circle diagram, evaluates all six trigonometric functions, computes arc length and sector area for any radius, and models uniform circular motion (angular velocity, linear velocity, centripetal acceleration). Every result is computed in your browser using IEEE 754 double-precision arithmetic — no data is uploaded anywhere.

How it works

The core conversion is two lines of algebra:

• radians = degrees x pi / 180
• degrees = radians x 180 / pi

The value of pi used is JavaScript’s Math.PI, which equals 3.141592653589793 — the nearest 64-bit double to the true irrational value.

Unit circle. Any angle theta maps to a point (cos theta, sin theta) on the circle of radius 1. The diagram draws the radius to that point and overlays dotted lines to each axis so you can read off the cosine (horizontal) and sine (vertical) values directly.

All six trig functions. Once theta is known in radians, the six standard functions follow from the definitions: sin, cos, tan = sin/cos, then their reciprocals csc = 1/sin, sec = 1/cos, cot = 1/tan. The Pythagorean identity sin²θ + cos²θ = 1 is verified live to show floating-point precision (you should see a value indistinguishable from 1).

Arc length and sector area. For a circle of radius r with central angle theta (radians), the arc length is s = r x |theta| and the sector area is A = (1/2) x r² x |theta|. These come directly from the definition of a radian: one radian sweeps an arc of length r, so theta radians sweep r x theta.

Angular motion. In uniform circular motion an object rotates through angle theta in time t, giving angular velocity omega = theta / t (rad/s). The tangential speed is v = omega x r and the centripetal (inward) acceleration is a = omega² x r = v² / r (m/s²). Centripetal force on a mass m is then F = m x omega² x r (newtons).

Worked example

Convert 120 degrees and explore the geometry with radius r = 10 m in t = 3 s.

Step 1 — Convert: radians = 120 x pi / 180 = 2pi/3 ≈ 2.094395 rad. The exact pi fraction is 2pi/3.

Step 2 — Trig values at 120 degrees:

Step 3 — Arc and sector (r = 10 m):

• Arc length s = 10 x 2.094395 ≈ 20.944 m
• Sector area A = 0.5 x 100 x 2.094395 ≈ 104.720 m²
• Full circumference for comparison: 2 x pi x 10 ≈ 62.832 m

Step 4 — Angular motion (r = 10 m, t = 3 s):

• omega = 2.094395 / 3 ≈ 0.698 rad/s
• Linear velocity v = 0.698 x 10 ≈ 6.98 m/s
• Centripetal acceleration a = 0.698² x 10 ≈ 4.87 m/s²

The angle 120° sits in Quadrant II (between 90° and 180°), where sine is positive and cosine is negative — confirming the table above.

Formula reference

All standard physical constants used in the Angular Motion tab: g = 9.81 m/s² (standard gravity, for context). No network calls are made; all constants are hardcoded to their NIST/CODATA values inside the browser.