Radians to Degrees: Complete Guide, Formula, π Fractions, Unit Circle, Trigonometry, and Why 1 Radian ≈ 57.3°
Converting radians to degrees reverses the natural angle measurement back into the familiar 360-degree system. While radians are the mathematical standard, degrees remain dominant in everyday life — compasses, protractors, architecture, and navigation all use degrees. This converter supports both decimal radians and π-notation input, providing exact degree values, animated unit circle visualization, and complete trigonometric breakdowns.
The Formula
To convert radians to degrees, multiply by 180/π. This works because π radians = 180°, so multiplying by 180/π cancels the π. The factor 180/π ≈ 57.2958 converts any radian value to degrees.
Key values:
0 = 0° | π/6 = 30° | π/4 = 45°
π/3 = 60° | π/2 = 90° | 2π/3 = 120°
3π/4 = 135° | 5π/6 = 150° | π = 180°
3π/2 = 270° | 2π = 360°
1 radian = 180/π ≈ 57.2958°
Understanding 1 Radian
One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Since a circle's circumference is 2πr, a full circle contains 2π radians. Therefore 1 radian = 360°/(2π) = 180°/π ≈ 57.2958°. This seemingly awkward number is actually the most natural angle unit — it makes the arc length formula simply s = rθ, with no conversion factors needed.
π Fraction Inputs
This converter accepts π-notation: type "pi/4" for π/4, "2pi/3" for 2π/3, "pi" for π, or "3pi" for 3π. This is how radians naturally appear in mathematics — as multiples or fractions of π. Common angles that produce integer degrees are always π fractions: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 2π = 360°.
π/12 ≈ 0.2618 rad = 15°
π/6 ≈ 0.5236 rad = 30°
π/4 ≈ 0.7854 rad = 45°
π/3 ≈ 1.0472 rad = 60°
π/2 ≈ 1.5708 rad = 90°
π ≈ 3.1416 rad = 180°
3π/2 ≈ 4.7124 rad = 270°
2π ≈ 6.2832 rad = 360°
1 rad ≈ 57.2958°
Where You Encounter Radians
Radians appear in many contexts that require conversion back to degrees for interpretation. Programming languages return values in radians from Math.atan2() — converting to degrees makes results human-readable. Physics angular measurements (gyroscope data, rotational velocity) are in rad/s — converting to °/s aids understanding. Signal processing phase values are in radians — converting to degrees clarifies phase relationships. Robotics joint angles from inverse kinematics arrive in radians but servo commands may need degrees.
π/4 = 45° (divide 180 by 4)
π/3 = 60° (divide 180 by 3)
π/2 = 90° (divide 180 by 2)
For nπ/m: degrees = n × 180 / m
Quick check: 1 rad ≈ 57.3°, so 2 rad ≈ 114.6°, 3 rad ≈ 171.9°.
Special Angles and Trig Values
The "special angles" that appear throughout mathematics have memorable sine and cosine values: 0 rad (0°): sin=0, cos=1; π/6 (30°): sin=½, cos=√3/2; π/4 (45°): sin=cos=√2/2; π/3 (60°): sin=√3/2, cos=½; π/2 (90°): sin=1, cos=0. These repeat through all four quadrants with appropriate sign changes. Knowing these by heart — in both radians and degrees — is essential for trigonometry, physics, and engineering.
How to Use This Converter
Enter radians in the input field — either as a decimal (like 1.5708) or using π notation (like "pi/2", "2pi/3", "pi"). Press "Convert" for the full breakdown: animated unit circle, exact degree value, all angle units, sin/cos values, and reference chart. Quick-value buttons provide standard unit circle angles. To convert degrees to radians, use the switch link or visit our Degrees to Radians converter.