Angle Converter

What is an angle?

An angle measures the rotation between two rays or the opening between two lines. We meet angles everywhere: the 90° corner of a room, the 30° slope of a ramp, the π/2 radians in a quarter turn, the bearing of a ship, the field of view of a camera. The most common units are degrees and radians, but gradians, arcminutes, arcseconds, and turns each serve specific needs. An angle converter ties them together through a single base.

Degrees vs radians vs gradians

A degree splits a circle into 360 parts — a convention from ancient Babylonian astronomy. A radian is the angle whose arc equals the radius; a full circle is 2π radians (≈ 6.283). Radians are the natural unit in calculus and physics because derivatives of sine and cosine are cleanest there. A gradian splits a circle into 400 parts (100 gradians per right angle), handy in surveying. The turn simply counts whole rotations. Our tool converts among all of them exactly.

The base-unit method

We convert every value into degrees (the base) using each unit's degree factor, then divide by the target factor. For radians the factor is 180/π, so the conversion stays mathematically exact. For example, 1 radian = 57.29578° and 1° = 0.0174533 rad. 1 gradian = 0.9°, 1 turn = 360°, 1 arcminute = 1/60°, 1 arcsecond = 1/3600°. The tool computes any pair instantly.

Key conversion factors (to degrees)

• 1 degree = 1° (base)
• 1 radian = 180/π ≈ 57.29578°
• 1 gradian = 0.9°
• 1 minute of arc = 1/60° ≈ 0.0166667°
• 1 second of arc = 1/3600° ≈ 0.0002778°
• 1 turn = 360°

Degrees to radians (the key formula)

The relationship you will use most: radians = degrees × π / 180. So 180° = π rad, 90° = π/2 rad ≈ 1.5708 rad, and 45° = π/4 ≈ 0.7854 rad. Going the other way, degrees = radians × 180/π. Since π is irrational, the result is rounded only at display (six decimals); the underlying factor is exact.

Arcminutes and arcseconds

For very small angles — astronomy, optics, firearms, and geodetic surveying — degrees are too coarse, so we subdivide: 1° = 60 arcminutes (′), and 1 arcminute = 60 arcseconds (″). The full moon is about 30 arcminutes (½°) wide; a GPS position might be accurate to a few arcseconds. A parsec is defined by a 1-arcsecond parallax. Our converter handles these tiny units precisely so you can express a fraction of a degree as arcseconds without mental arithmetic.

Real-world examples

A right angle is 90° = π/2 rad = 100 gradians = 0.25 turn. A gentle 5° road grade = 0.087266 rad = 300 arcminutes. A camera's 78° field of view = 1.3614 rad. A machinist's 0.5° tolerance = 30 arcminutes = 1800 arcseconds. A propeller spinning at 2 turns/s rotates 720°/s = 4π rad/s. Whether you think in slopes, spins, or sky positions, the converter aligns the units.

Common mistakes

The biggest error is mixing degrees and radians in a formula — many calculators default to radians, so entering 90 expecting "1" gives 1.5708 instead. Always check the mode. Another is forgetting that gradians are not degrees (a 100-grad right angle vs 90°). A third is confusing arcminutes/arcseconds (angles) with minutes/seconds of time (used in astronomy for right ascension, where 1 hour = 15°). Use the converter to keep the unit unambiguous and exact.

Frequently asked questions