Degrees to Radians: Complete Guide, Formula, Unit Circle, Trigonometry, and Why Radians Matter
Degrees and radians are two ways to measure angles. Degrees divide a full circle into 360 equal parts — a convention inherited from ancient Babylonian astronomy. Radians measure angles by the ratio of arc length to radius — one radian is the angle where the arc length equals the radius. Converting between them is fundamental to trigonometry, calculus, physics, and engineering.
The Formula
To convert degrees to radians, multiply by π/180. This works because a full circle is 360° = 2π radians, so 1° = 2π/360 = π/180 radians. The factor π/180 ≈ 0.0174533 converts any degree value to radians.
Key values:
0° = 0 rad | 30° = π/6 | 45° = π/4
60° = π/3 | 90° = π/2 | 120° = 2π/3
135° = 3π/4 | 150° = 5π/6 | 180° = π
270° = 3π/2 | 360° = 2π
Why Radians?
Radians simplify mathematics profoundly. In calculus, d/dx sin(x) = cos(x) — but only when x is in radians. With degrees, an ugly factor of π/180 appears in every derivative. The arc length formula s = rθ works directly when θ is in radians. Angular velocity in physics (ω = θ/t) requires radians. Taylor series for sin and cos assume radians. Essentially, radians are the "natural" unit for angles — degrees are a human convenience.
The Unit Circle
The unit circle (radius = 1) is the foundation of trigonometry. Every point on the circle has coordinates (cos θ, sin θ) where θ is measured in radians from the positive x-axis. The key angles and their sine/cosine values: 0° (0 rad): sin=0, cos=1; 30° (π/6): sin=½, cos=√3/2; 45° (π/4): sin=√2/2, cos=√2/2; 60° (π/3): sin=√3/2, cos=½; 90° (π/2): sin=1, cos=0. These values repeat with sign changes through all four quadrants.
15° = π/12 ≈ 0.2618 rad
30° = π/6 ≈ 0.5236 rad
45° = π/4 ≈ 0.7854 rad
60° = π/3 ≈ 1.0472 rad
90° = π/2 ≈ 1.5708 rad
120° = 2π/3 ≈ 2.0944 rad
180° = π ≈ 3.1416 rad
270° = 3π/2 ≈ 4.7124 rad
360° = 2π ≈ 6.2832 rad
Applications
Radians appear everywhere in STEM: Physics — angular velocity (rad/s), oscillation frequency, wave equations. Engineering — servo motor positions, robotics joint angles, signal processing (phase in radians). Computer graphics — rotation matrices, shader math (GPU functions use radians). Navigation — latitude/longitude calculations use radians internally. Programming — Math.sin(), Math.cos() in JavaScript, Python, C++, etc. all expect radians.
45° = π/4 (45/180 = 1/4)
60° = π/3 (60/180 = 1/3)
90° = π/2 (90/180 = 1/2)
For any angle: divide degrees by 180, multiply by π.
Quick check: 1 radian ≈ 57.3° (since 180/π ≈ 57.296).
Exact π Fractions
Many common angles produce clean π fractions. Multiples of 15° always yield π fractions with small denominators: 15° = π/12, 30° = π/6, 45° = π/4, 60° = π/3, 75° = 5π/12, 90° = π/2. For angles that are not multiples of 15°, the radian value is typically expressed as a decimal. For example, 1° = π/180 ≈ 0.01745 rad — not a "clean" fraction.
How to Use This Converter
Enter degrees in the input field and press "Convert" for the full breakdown: animated unit circle with ray and arc, exact π fraction when possible, all angle units (radians, gradians, turns), sin/cos values, step-by-step calculation, and a comprehensive reference chart. Use the quick-value buttons for common angles. To convert in the reverse direction (radians → degrees), use the switch link or visit our Radians to Degrees converter.