CALTRX

Radian to Degree Converter

Convert angles between degrees and radians for trigonometry and engineering. Handles both directions with full precision.

Enter your values above to see the results.

Tips & Notes

  • ✓Multiply degrees by π/180 to get radians. Multiply radians by 180/π to get degrees.
  • ✓Memorize key conversions: 180°=π, 90°=π/2, 45°=π/4, 30°=π/6, 60°=π/3.
  • ✓Full rotation: 360°=2π radians. Half: 180°=π. Quarter: 90°=π/2.
  • ✓Arc length = r×θ where θ is in radians. For r=5, θ=π/3: arc = 5π/3 ≈ 5.24 units.
  • ✓Calculator: always check degree/radian mode before computing sin, cos, tan.

Common Mistakes

  • ✗Multiplying by π/180 when converting from radians — should multiply by 180/π.
  • ✗Forgetting that π ≈ 3.14159, not 180. π is the radian measure of 180°, not the number 180.
  • ✗Leaving calculator in wrong mode: sin(90) in radians ≈ 0.894, not 1. Always check mode.
  • ✗Rounding π prematurely. Use full precision (3.14159265...) for accurate conversions.
  • ✗Confusing degrees-minutes-seconds (DMS) with decimal degrees. 30°30 = 30.5°, not 3030.

Radian to Degree Converter Overview

Radians and degrees are two units for measuring angles — both describe the same physical rotation, but using different scales. Degrees divide a full rotation into 360 equal parts, a choice tracing back to ancient Babylonian astronomy and the approximate 360-day year. Radians measure angles by the arc length they subtend on a unit circle — one radian is the angle corresponding to an arc length exactly equal to the radius. While degrees are intuitive for everyday use, radians are mathematically natural: every calculus formula involving trigonometry requires angles in radians to work without extra conversion factors.

Conversion formulas:

Radians = Degrees × π/180
Degrees = Radians × 180/π
EX: 45° → 45 × π/180 = π/4 ≈ 0.7854 rad | EX: π/3 rad → π/3 × 180/π = 60°
EX: 270° → 270 × π/180 = 3π/2 ≈ 4.7124 rad | EX: 5π/6 rad → 5π/6 × 180/π = 150°
Radians are the mathematically natural unit for angle measurement — the arc length formula s = rθ works only in radians, meaning arc length equals radius times angle directly, without any conversion factor. Calculator angle mode: most scientific calculators have a DEG/RAD mode switch. sin(90) in DEG mode = 1.0. sin(90) in RAD mode ≈ 0.894 — wrong if you intended 90°. Always verify the mode before any trigonometric calculation; this is the single most common source of errors in physics and engineering.

Frequently Asked Questions

Radians measure angles as arc length divided by radius. In a full circle, the arc length equals the circumference = 2πr. Dividing by r gives 2π radians in a full circle. Therefore: 360° = 2π radians, 180° = π radians, 90° = π/2 radians. To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. Example: 45° × π/180 = π/4 radians ≈ 0.7854 rad. Example: 2.5 rad × 180/π ≈ 143.24°.

Calculus requires radians because the derivative of sin(x) equals cos(x) only when x is in radians. In degrees, d/dx[sin(x°)] = (π/180)cos(x°) — an extra conversion factor appears. Similarly, the Taylor series sin(x) = x − x³/6 + x⁵/120 − ... is only valid with x in radians. All calculus formulas, differential equations, and Taylor expansions involving trigonometric functions assume radian input. Using degrees in these formulas produces wrong answers.

Key radian values to memorize: 0° = 0, 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. These correspond to the standard angles on the unit circle where sine and cosine have exact values. Example: sin(π/6) = 1/2, cos(π/3) = 1/2, tan(π/4) = 1, sin(π/2) = 1. Memorizing these eliminates conversions for the most common angles.

Arc length s = r × θ where θ is in radians. Example: a circle with radius 5 cm, central angle 60° = π/3 radians. Arc length = 5 × π/3 = 5π/3 ≈ 5.24 cm. Sector area = (1/2) × r² × θ = (1/2) × 25 × π/3 = 25π/6 ≈ 13.09 cm². These formulas require radians — using degrees without conversion gives incorrect results. Angular velocity ω is measured in radians per second: a wheel rotating at 300 RPM = 300 × 2π/60 = 10π ≈ 31.4 rad/s.

Navigation and surveying use degrees, minutes, and seconds (DMS). 1° = 60 arcminutes ('), 1' = 60 arcseconds (''). Example: 37°30'45'' = 37 + 30/60 + 45/3600 = 37.5125°. GPS coordinates use decimal degrees: 37.5125°. To convert to radians: 37.5125 × π/180 = 0.6549 rad. Star positions and telescope pointing use DMS; physics and engineering calculations use radians; navigation displays use decimal degrees.

Angular frequency ω (omega) in physics equals 2πf where f is frequency in Hz. A 60 Hz electrical system: ω = 2π × 60 = 120π ≈ 377 rad/s. Voltage oscillates as V(t) = Vmax × sin(ωt). A pendulum of length L has angular frequency ω = √(g/L) radians per second, giving period T = 2π/ω = 2π√(L/g). A 1-meter pendulum: T = 2π√(1/9.81) ≈ 2.006 seconds. Radians make these formulas clean and dimensionally consistent.