Angle Converter

Three Very Different Ways to Slice a Circle

Humans have invented at least three distinct systems for measuring angles, each rooted in a different mathematical or practical tradition. Degrees divide a full circle into 360 parts — a number chosen by Babylonian astronomers roughly 4,000 years ago, likely because 360 has many divisors (2, 3, 4, 5, 6, 8, 9, 10, 12…) making it easy to work with in pre-calculator arithmetic. Radians are the natural mathematical unit, defining angle by arc length: one radian is the angle subtended when the arc length equals the radius, making a full circle exactly 2π radians. Gradians (also called gon or grad) divide a right angle into 100 parts, making a full circle 400 gradians — a decimal-friendly system introduced during the French Revolution alongside the metric system. All three remain in active use today, and this converter handles them all plus arcminutes, arcseconds, and turns.

Why Radians Matter in Mathematics and Programming

If you have ever written code using Math.sin(), Math.cos(), or Math.atan2(), you have already used radians — every programming language and scientific calculator expects trigonometric function inputs in radians, not degrees. The reason is mathematical elegance: in radian measure, the derivative of sin(x) is simply cos(x), with no conversion factor. In degree measure, the derivative of sin(x°) involves π/180 everywhere, cluttering every calculus expression. This is why physics, signal processing, Fourier analysis, and computer graphics all work internally in radians. The conversion is straightforward — multiply degrees by π/180 to get radians, divide by π/180 to go back — but it trips up programmers and students constantly. Use this converter to check your inputs before passing them to trigonometric functions.

Arcminutes and Arcseconds: Precision Navigation and Astronomy

One degree can be subdivided into 60 arcminutes ('), and each arcminute into 60 arcseconds (''). This sexagesimal system mirrors the way degrees themselves were defined, and it remains the standard for geographic coordinates, celestial positioning, and precision optics. The International Space Station orbits at roughly 51.6° inclination. The coordinates of the Eiffel Tower are 48°51′29.6″N, 2°17′40.2″E — read as 48 degrees, 51 arcminutes, 29.6 arcseconds north. In astronomy, stellar positions are recorded to milliarcsecond precision using very long baseline interferometry. In rifle optics, scope adjustments are made in MOA (minute of angle) — one arcminute corresponds to approximately 1 inch at 100 yards, a fact every precision shooter knows. For distances that accompany angular measurements in navigation, the Length Converter covers nautical miles, metres, and feet.

Gradians: The Metric Angle Unit That Never Quite Won

The gradian (grad or gon) divides a right angle into 100 equal parts, so a full circle = 400 grad and a straight line = 200 grad. The appeal is obvious: a 45° angle is exactly 50 grad; a 30° angle is 33.333... grad. It was designed to fit neatly into decimal arithmetic, and was adopted in France, Switzerland, and parts of Central Europe for surveying and civil engineering. Modern surveying instruments from European manufacturers often display angles in gradians, and the unit persists in land survey records from these regions. Despite never achieving global adoption, the gradian still appears regularly enough that any tool dealing with survey data, GIS shapefiles, or European engineering drawings needs to handle it. The calculator button labelled "DRG" on scientific calculators cycles through Degrees, Radians, and Gradians — an acknowledgement that all three remain in active use.

The Turn: One Full Rotation as a Single Unit

The turn (also called revolution, full circle, or cycle) represents one complete 360° rotation as the value 1. It is useful when counting rotations — a motor spinning at 3,000 RPM completes 3,000 turns per minute, a figure that translates to 3,000 × 2π = 18,849.6 radians per minute or 3,000 × 360 = 1,080,000 degrees per minute. The turn is also the natural unit for phase angles in signal processing: a full cycle of a sine wave is 1 turn, making it intuitive for describing phase shifts. In computer graphics and game engines, quaternion rotations are sometimes expressed as fractions of a full turn. For speed measurements involving rotational motion, the Speed Converter covers linear velocity units that complement angular velocity.

Quick Angle Conversion Reference

• 1 full circle = 360° = 2π rad ≈ 6.28318 rad = 400 grad = 1 turn
• 1 radian = ≈57.2958° = 63.6620 grad = 3,437.75 arcmin
• 1° = π/180 rad ≈ 0.017453 rad = 1.11111 grad = 60 arcmin = 3,600 arcsec
• 1 arcminute = 1/60° = 0.000291 rad = 0.018519 grad
• 1 arcsecond = 1/3600° = 4.848 × 10⁻⁶ rad
• Right angle = 90° = π/2 rad = 100 grad