Angle Converter
Convert angles between degrees, radians, gradians, arcminutes, arcseconds, and turns.
How to use this tool
- Enter angle and from unit in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your radians and the full breakdown beneath it.
Formula
All angles are first converted to degrees, then converted to each output unit:
degrees = value × factorfrom
Factors to degrees: 1 radian = 180 ÷ π ≈ 57.2958°, 1 gradian = 0.9°, 1 arcminute = 1/60°, 1 arcsecond = 1/3600°, 1 turn = 360°.
From degrees: radians = degrees × π ÷ 180 | gradians = degrees ÷ 0.9 | arcminutes = degrees × 60 | arcseconds = degrees × 3600 | turns = degrees ÷ 360.
How it works
The converter uses degrees as its internal pivot. The input is multiplied by the appropriate factor to obtain degrees, and all output values are then derived from that degree measure using standard geometric relationships: 2π radians and 400 gradians both represent one full rotation, as does 1 turn or 360 degrees.
Common mistake: Passing degrees to a trigonometric function that expects radians (or vice versa). Most programming languages and scientific calculators use radians by default — entering 180 where radians are expected gives sin(180 rad) ≈ −0.8012, not sin(180°) = 0.
Worked example
Convert 180 degrees to all angle units
- Input: 180 degrees. Degrees is the pivot, so degrees = 180.
- Radians = 180 × π / 180 = π ≈ 3.14159.
- Gradians = 180 / 0.9 = 200.
- Arcminutes = 180 × 60 = 10,800; Arcseconds = 180 × 3,600 = 648,000; Turns = 180 / 360 = 0.5.
180° = 3.14159 rad = 200 grad = 10,800 arcminutes = 648,000 arcseconds = 0.5 turns.
Common mistakes to avoid
- Entering degrees in a formula that expects radians -- trigonometric functions in most programming languages (sin, cos, tan) take radians, so sin(90) returns sin(90 rad), not sin(90 degrees).
- Confusing gradians with degrees -- 400 gradians = 360 degrees = a full circle; a right angle is 100 gradians, not 90.
- Forgetting that arcminutes and arcseconds are subdivisions of a degree (1 degree = 60 arcminutes = 3,600 arcseconds), not of a minute of time.
Key terms
- Radian (rad)
- 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 circle is 2π ≈ 6.2832 radians.
- Gradian (grad)
- Also called a gon. A unit where a right angle equals exactly 100 gradians and a full circle equals 400. Mainly used in surveying and some European engineering contexts.
- Arcminute (′)
- One sixtieth of a degree (1° = 60′). Used in navigation, astronomy, and optometry. Not to be confused with minutes of time.
- Arcsecond (″)
- One sixtieth of an arcminute, or 1/3,600 of a degree. Used in precise contexts such as astronomical coordinates and geodetic measurements.
Frequently asked questions
- How many radians are in a full circle?
- A full circle is 2*pi radians (approximately 6.2832 rad). A half circle (straight line) is pi radians (approximately 3.1416 rad), which equals 180 degrees.
- What is a gradian and who uses it?
- A gradian (also called gon or grade) divides a right angle into 100 equal parts, making a full circle 400 gradians. It is used in surveying, especially in continental Europe, because slope percentages and gradian angles align conveniently.
- What are arcminutes used for?
- Arcminutes (1/60 of a degree) and arcseconds (1/3600 of a degree) are used in astronomy, GPS coordinates, and optical precision work. A GPS coordinate like 48 degrees 51 arcminutes 24 arcseconds N pinpoints a location to about 30 metres.