Angle Converter – Degrees, Radians, Gradians and Arcminutes

Degrees and radians

A full circle is 360 degrees or 2π radians, so the conversion formulas are:

• radians = degrees × π ÷ 180
• degrees = radians × 180 ÷ π

One radian is the angle at which the arc length equals the radius, about 57.2958°. Common values: 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 90° = π/2 ≈ 1.5708 rad and 180° = π ≈ 3.1416 rad.

Why radians?

Radians are the natural unit for mathematics and physics. Calculus formulas, such as the derivative of sin x being cos x, only hold when x is in radians. Most programming languages – JavaScript, Python, PHP and spreadsheet functions such as SIN and COS – also expect radians. Convert degrees to radians before calling those functions, or use a built-in helper such as RADIANS() in spreadsheets. On a calculator, the DEG and RAD modes decide which unit is used; being in the wrong mode is a common cause of wrong answers in exams.

Gradians

The gradian (also gon or grade) divides a right angle into 100 parts, so a full circle is 400 gradians. It is used in surveying and civil engineering in some European countries because slopes and right angles become round numbers. 100 gradians equal 90 degrees, and one gradian is 0.9°.

Arcminutes and arcseconds

For very small angles, a degree is divided into 60 arcminutes (′), and an arcminute into 60 arcseconds (″). They are used in astronomy, navigation, optics and for the accuracy of rifle scopes. The full Moon is about 31 arcminutes wide, and one arcminute of latitude corresponds to one nautical mile on the Earth’s surface.

Slopes and angles

A slope given as a percentage is not an angle: a 100% slope is 45°. To convert, take the arctangent of the percentage divided by 100; a 10% road gradient is about 5.7°.

Sources

Conversion factors follow the SI definitions in the BIPM SI Brochure and the exact factors for US customary and imperial units in NIST Special Publication 811.