Convert Degree (°) to Radian (rad) instantly. Enter any value and get the result immediately.
° → rad Converter
| Degree (°) | Radian (rad) |
|---|---|
| 0.1 ° | 0.00174533 rad |
| 0.5 ° | 0.00872664 rad |
| 1 ° | 0.01745329 rad |
| 2 ° | 0.03490657 rad |
| 5 ° | 0.08726643 rad |
| 10 ° | 0.17453286 rad |
| 20 ° | 0.34906573 rad |
| 50 ° | 0.87266431 rad |
| 100 ° | 1.74532863 rad |
| 200 ° | 3.49065726 rad |
| 500 ° | 8.72664314 rad |
| 1000 ° | 17.45328628 rad |
| 5000 ° | 87.2664314 rad |
| 10000 ° | 174.53286279 rad |
Converting degrees to radians is one of the most essential skills in mathematics, physics, and engineering. Degrees are the familiar everyday unit for angles, but radians are the natural unit of angular measurement used in calculus, trigonometry functions in programming, rotational physics, and all higher-level mathematics. To convert, multiply the degree value by π/180 (approximately 0.01745329). Use the converter above for instant results, or follow the formula and examples below.
Step-by-step example — Convert 90° to radians:
Step-by-step example — Convert 45° to radians:
Step-by-step example — Convert 270° to radians:
Degree (°) is the most widely used unit of angular measurement in everyday life. A full circle is divided into 360 equal degrees — a convention inherited from ancient Babylonian astronomers who used a base-60 number system and noted that the Earth completes roughly 360 days in a year. One degree represents 1/360th of a full rotation. Degrees are intuitive and human-friendly: a right angle is 90°, a straight line is 180°, and a full turn is 360°. They are the standard unit in navigation, geography, architecture, and school-level geometry worldwide.
Radian (rad) is the SI standard unit of angular measurement and the natural unit of angle in mathematics. One radian is defined as the angle subtended at the center of a circle by an arc whose length equals the radius of that circle. Because the circumference of a circle is 2π times its radius, a full circle contains exactly 2π radians (approximately 6.2832 rad). This geometric definition makes radians uniquely suited to calculus and physics: the derivatives of trigonometric functions (sin, cos, tan) are clean and simple only when angles are expressed in radians. In programming languages and scientific libraries — including Python's math module, C's math.h, MATLAB, and JavaScript's Math object — all trigonometric functions accept and return angles in radians by default. A radian is a larger unit than a degree: 1 rad ≈ 57.2958°.
| Degrees (°) | Radians (exact) | Radians (decimal) | Common Reference |
|---|---|---|---|
| 0° | 0 | 0 rad | No rotation |
| 30° | π/6 | 0.5236 rad | Interior angle of equilateral triangle ÷ 2 |
| 45° | π/4 | 0.7854 rad | Half a right angle — diagonal |
| 60° | π/3 | 1.0472 rad | Interior angle of equilateral triangle |
| 90° | π/2 | 1.5708 rad | Right angle — quarter circle |
| 120° | 2π/3 | 2.0944 rad | Interior angle of regular hexagon |
| 135° | 3π/4 | 2.3562 rad | Interior angle of regular octagon |
| 180° | π | 3.1416 rad | Straight angle — half circle |
| 270° | 3π/2 | 4.7124 rad | Three-quarter rotation |
| 360° | 2π | 6.2832 rad | Full circle — one complete rotation |
The formula is: rad = ° × (π ÷ 180). Multiply any degree value by π/180 (approximately 0.01745329) to get the equivalent angle in radians. The exact form is always preferred in mathematics: for example, 180° = π rad exactly.
180° = π radians (approximately 3.14159 rad). This is the single most important degree-radian relationship to memorize — it is the foundation of all degree-to-radian conversions and appears constantly in trigonometry, calculus, and physics.
360° = 2π radians (approximately 6.28318 rad). A full circle contains exactly 2π radians — a direct consequence of the radian's definition, since the circumference of a unit circle (radius = 1) is 2π.
90° = π/2 radians (approximately 1.5708 rad). A right angle is exactly π/2 rad — one of the key benchmark values in trigonometry and the argument at which sin(x) reaches its maximum value of 1.
Mathematicians prefer radians because they make calculus and analysis dramatically simpler. With radians, the derivative of sin(x) is cos(x) — clean and elegant. With degrees, the derivative of sin(x°) becomes (π/180)cos(x°), introducing an ugly constant into every formula. Radians also arise naturally from the geometry of circles: they represent a direct ratio of arc length to radius, making them a dimensionless, physically meaningful unit rather than an arbitrary subdivision of the circle.
1 radian = 180/π degrees ≈ 57.2958°. One radian is a surprisingly large angle — just over 57 degrees — because it corresponds to the angle at which the arc length equals the radius of the circle.
Use the key benchmark values: 180° = π rad. So divide your degree value by 180 and multiply by π. For common angles: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 120° = 2π/3, 180° = π, 270° = 3π/2, 360° = 2π. Memorizing these seven benchmarks covers virtually every angle encountered in school and university mathematics.
All major programming languages and scientific libraries use radians for trigonometric functions by default. Python's math.sin(), JavaScript's Math.sin(), C's sin(), Java's Math.sin(), and MATLAB's sin() all expect radian input. Always convert degrees to radians using the formula rad = degrees * Math.PI / 180 before passing angle values to these functions.