Angle Converter

An angle is the amount of turn between two rays (lines) that share a common endpoint called the vertex. It measures rotation or the opening between the two rays, and is expressed in units such as degrees, radians or gradians.

A degree (°) is a unit of angle equal to 1/360 of a full turn. The 360-degree circle dates back to ancient Babylonian astronomy. A right angle is 90°, a straight line is 180° and a full rotation is 360°.

A radian (rad) is the SI unit of angle. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full turn is 2π radians (≈ 6.2832), so 180° = π radians ≈ 3.14159 rad.

A gradian (gon or grade) divides a right angle into 100 parts, so a full circle is 400 gradians. It is used mainly in surveying and some European engineering because it makes percentage-style slope calculations convenient.

Multiply the number of degrees by π/180. For example, 180° × π/180 = π ≈ 3.14159 rad, and 90° × π/180 = π/2 ≈ 1.5708 rad. To go back, multiply radians by 180/π.

180 degrees equals exactly π radians, which is approximately 3.14159265 radians.

π radians equals exactly 180 degrees. Therefore 2π radians = 360°, π/2 radians = 90°, and 1 radian ≈ 57.2958°.

A right angle measures exactly 90 degrees (π/2 radians, or 100 gradians). It is the angle between two perpendicular lines, such as the corner of a square.

Radians make calculus and trigonometry simpler: the derivative of sin(x) is cos(x) only when x is in radians, and arc length is simply radius × angle. Because a radian is defined purely by the circle’s geometry, formulas avoid extra conversion factors like π/180.

Engineers and physicists use radians because angular velocity, angular acceleration, oscillation and wave equations come out cleanly in radians. For example, angular speed ω in rad/s relates directly to linear speed by v = ωr without any conversion constant.

A milliradian is one thousandth of a radian. It is widely used in optics, ballistics and rifle scopes: at 100 metres, 1 mrad subtends 10 cm, making range and adjustment estimates easy.

A military mil is an angular unit defined so that a full circle is 6400 mils (the NATO standard). It approximates a milliradian and is used for artillery and targeting because one mil subtends roughly one metre at 1000 metres. Other standards use 6000 (former Warsaw Pact) or 6300 (Sweden) mils per turn.

A bearing describes a direction as an angle. An azimuth (whole-circle) bearing is measured clockwise from north, from 0° to 360°. A quadrant bearing is written as N45°E or S30°W, measured east or west from the north or south line. This tool converts between the two.

An azimuth is always measured clockwise from north on a 0–360° scale. A (quadrant) bearing is measured from either north or south, toward east or west, and never exceeds 90°. For example, an azimuth of 225° is the same direction as the quadrant bearing S45°W.

The grade (slope percentage) equals the tangent of the angle times 100. So a 45° slope is tan(45°) × 100 = 100%, and a 100% grade is a 45° angle. A 10° slope is about 17.6%.

A compass point is 1/32 of a full turn, equal to 11.25°. The traditional mariner’s compass has 32 points — N, NbE, NNE, NEbN, NE, and so on — used for naming directions before precise degree bearings became common.

To normalise an angle, add or subtract full turns (360°, 2π rad or 400 gon) until it falls within one rotation. For example, 450° normalises to 90°, and −90° normalises to 270°. This tool does it automatically.

An arcminute (′) is 1/60 of a degree, and an arcsecond (″) is 1/60 of an arcminute, or 1/3600 of a degree. They measure very small angles in astronomy, navigation and surveying.

One degree equals 10/9 gradians (about 1.1111 gon). Conversely, one gradian equals 0.9 degrees. A full circle is 360° = 400 gon.

A revolution (or turn) is one complete rotation: 360°, 2π radians, 400 gradians or 6400 NATO mils. It is the natural unit for rotational speed, such as revolutions per minute (RPM).

Take the whole degrees, multiply the decimal part by 60 to get arcminutes, then multiply that remainder by 60 for arcseconds. For example, 12.5° = 12° 30′ 0″. Use the arcminute and arcsecond units to see the breakdown.

In astronomy the sky rotates 360° in 24 hours, so one hour of right ascension equals 15° of arc. The hour angle expresses position along the celestial equator in hours, minutes and seconds of time rather than degrees.

Conversions use exact factors based on π and are computed with double-precision arithmetic — accurate to about 15 significant figures. You can display from 0 to 15 decimal places, set a custom precision, or switch to scientific notation.

For a given angle, sine, cosine and tangent are ratios from a right triangle (or coordinates on the unit circle). For example sin(30°) = 0.5, cos(60°) = 0.5 and tan(45°) = 1. The tangent is undefined at 90° and 270°, where the cosine is zero.

Yes. In the value box you can type expressions like pi, 2pi or pi/2 (π is also accepted). The converter evaluates them, so pi/2 radians correctly converts to 90 degrees.

Yes. It is completely free with no sign-up, runs entirely in your browser so no data is uploaded, and works offline once loaded. Your history and favourites are stored only on your device.

A reflex angle is greater than 180° but less than 360°. For example, 200° and 300° are reflex angles. Angles below 90° are acute, exactly 90° is a right angle, between 90° and 180° is obtuse, and exactly 180° is a straight angle.

One radian equals 180/π degrees, which is approximately 57.29578 degrees.

A binary degree, or binary radian (brad), divides a full turn into 256 parts so a whole rotation fits in a single byte. It is used in computer graphics and embedded systems for compact angle storage; 1 brad = 1.40625°.