Angle Converter
An angle measures the amount of rotation between two rays that share a common endpoint, or the opening between two intersecting lines or surfaces. It is a fundamental quantity in geometry, trigonometry, navigation, engineering, and astronomy, and it can be expressed in several different units depending on the field of work.
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People convert between angle units because each discipline favors a different scale. Mathematicians and physicists rely on radians, surveyors and everyday users prefer degrees, some engineering and surveying traditions use gradians, and astronomers and opticians work with arcminutes and arcseconds to describe very small angles precisely.
Angle Units Explained
The radian (rad) is the SI unit of angle: one radian is the angle subtended at the center of a circle by an arc equal in length to the radius, so a full circle is exactly 2π radians. The degree (deg) divides a full circle into 360 equal parts and is the most familiar everyday unit. The gradian (grad), also called the gon or grade, divides a full circle into 400 parts, making a right angle exactly 100 grad. For fine subdivisions of a degree, the arcminute (arcmin) is one-sixtieth of a degree and the arcsecond (arcsec) is one-sixtieth of an arcminute, so one degree contains 60 arcminutes and 3,600 arcseconds.
How to Convert Angle Units
The cleanest method is to convert through a single reference: the full circle equals 2π rad = 360 deg = 400 grad. Degrees are also split into arcminutes and arcseconds, where 1 deg = 60 arcmin = 3,600 arcsec. From these you get exact factors. To convert degrees to radians, multiply by π/180 (about 0.0174533); to go from radians to degrees, multiply by 180/π (about 57.2957795). For example, 90 deg = 90 x π/180 = π/2 rad ≈ 1.5708 rad. Degrees to gradians: multiply by 400/360, so 90 deg = 100 grad. Arcseconds to degrees: divide by 3,600, so 1,800 arcsec = 0.5 deg.
Common Uses
- Solving trigonometry and calculus problems in radians
- Surveying land and reading bearings in degrees or gradians
- Aiming telescopes and reporting star positions in arcseconds
- Setting cut angles in carpentry, machining, and CAD
- Calculating headings in aviation and marine navigation
- Specifying field of view and resolution in optics and cameras
Accuracy Tips
Always check whether a calculator or programming function expects radians or degrees, since trigonometric functions in most software default to radians. When converting through π, keep several decimal places (π ≈ 3.14159265) to avoid rounding errors. Degrees, arcminutes, and arcseconds use base 60, not 100, so do not treat 0.5 deg as 50 arcmin; it is 30 arcmin.
Frequently Asked Questions
How many degrees are in a radian?
One radian equals 180/π degrees, which is approximately 57.2957795 degrees. Conversely, one degree equals π/180 radians, about 0.0174533 radians.
What is the difference between a degree and a gradian?
A degree divides a full circle into 360 parts, while a gradian divides it into 400 parts. This makes a right angle 90 degrees but 100 gradians, and 1 gradian equals exactly 0.9 degrees.
How many arcseconds are in a degree?
There are 3,600 arcseconds in one degree, because each degree contains 60 arcminutes and each arcminute contains 60 arcseconds (60 x 60 = 3,600).
Why do mathematicians use radians instead of degrees?
Radians are based directly on the circle's radius and arc length, which makes calculus formulas for derivatives and integrals of trigonometric functions clean and free of extra conversion constants. A full circle is exactly 2π radians.
What is 1 radian in degrees, minutes, and seconds?
One radian is about 57.2957795 degrees, which works out to roughly 57 degrees, 17 arcminutes, and 45 arcseconds.