Angle Converter — degrees, radians and gradians

180 degrees equals exactly π radians, approximately 3.14159 rad. This is the straight angle (half revolution). Formula: radians = degrees × π / 180. For 180°: 180 × π / 180 = π ≈ 3.14159 rad.

Degree (°): full revolution = 360°. Used in everyday life, navigation and surveying. Radian (rad): full revolution = 2π rad ≈ 6.2832. The SI unit, fundamental in mathematics and physics — simplifies trigonometric formulas. Gradian (g or gon): full revolution = 400 g. Used mainly in European geodesy and military applications — one quadrant = 100 g.

Formula: radians = degrees × π / 180. Examples: 90° → π/2 ≈ 1.5708 rad; 45° → π/4 ≈ 0.7854 rad; 30° → π/6 ≈ 0.5236 rad; 360° → 2π ≈ 6.2832 rad. Remember: 1 degree = π/180 ≈ 0.01745 rad.

Formula: degrees = radians × 180 / π. Examples: π rad → 180°; π/2 rad → 90°; π/4 rad → 45°; 1 rad → 180/π ≈ 57.2958°. Remember: 1 radian is the angle at which the arc on a unit circle equals the radius.

Gradians (symbol g or gon) are used mainly in geodesy, cartography and military applications — particularly popular in France, Germany and Eastern Europe. One quadrant (right angle) = 100 g, simplifying field calculations. Full revolution = 400 g. In Poland encountered in precision geodesy and older military maps.

A radian is the SI unit of angle defined as the central angle subtending an arc equal to the radius. For a unit circle (r=1): angle in radians = arc length. Why important: trigonometric functions have simpler derivatives (sin'(x) = cos(x) in radians), physics formulas are more elegant (ω = dφ/dt), and calculus results are correct.

Gradians to degrees: degrees = gradians × 360 / 400 = gradians × 0.9. Gradians to radians: radians = gradians × π / 200. Examples: 100 g = 90° = π/2 rad; 200 g = 180° = π rad; 400 g = 360° = 2π rad. Remember: 1 gradian = 0.9° = π/200 rad.

0° = 0 rad; 30° = π/6 ≈ 0.524 rad; 45° = π/4 ≈ 0.785 rad; 60° = π/3 ≈ 1.047 rad; 90° = π/2 ≈ 1.571 rad; 120° = 2π/3 ≈ 2.094 rad; 180° = π ≈ 3.142 rad; 270° = 3π/2 ≈ 4.712 rad; 360° = 2π ≈ 6.283 rad. The angles π/6, π/4, π/3 are especially important in trigonometry.

Yes — negative angles are mathematically valid. A negative angle represents clockwise rotation (mathematical convention: positive = counterclockwise). Example: −90° = −π/2 rad = −100 g. The calculator handles values from −360,000 to 360,000 degrees.

The calculator uses JavaScript's built-in Math.PI constant, which approximates π to 15–17 significant digits: 3.141592653589793. Conversions: from degrees: rad = deg × Math.PI / 180, grad = deg × 400/360; from radians: deg = rad × 180 / Math.PI, grad = deg × 400/360; from gradians: deg = grad × 360/400, rad = deg × Math.PI / 180. Results are rounded to 6 decimal places.

In radians, the derivative of sin(x) is cos(x) with no extra factor. In degrees, d/dx sin(x°) = (π/180)·cos(x°) — an awkward constant. Similarly, arc length s = rθ (θ in radians) is elegantly simple. In physics, angular velocity ω = dθ/dt, frequency formulas and wave equations all assume radian measure. Switching to degrees would require π/180 factors everywhere.