Angle Converter
Angles arrive in three currencies: degrees for everyday geometry, radians for mathematics and physics, and gradians for surveying and some European tools. Add arcminutes and arcseconds — the units that let you quote 90° 0′ 0″ or a star's position — and the conversion stops being optional. A right angle is 90°, 1.5707963268 rad, 100 gon or a quarter of a turn, and this converter shows all of them together.
Enter values
Results
- In radians (rad)
- 1.5708
- In gradians (gon)
- 100
- In arcminutes (arcmin)
- 5,400
- In arcseconds (arcsec)
- 324,000
- In turns (turn)
- 0.25
- Exact factor
- 1 ° = 0.0174532925 rad
How is the Angle Converter calculated?
Enter an angle, choose the unit it is written in, and pick the unit you want back. The headline result shows your answer, and the rows beneath repeat the same angle in degrees, radians, gradians, arcminutes, arcseconds and turns so a slip of the decimal is obvious. The last row prints the exact factor — 1 rad = 57.29577951308232° — for checking by hand. Decimals are welcome: 12.5° and 0.75 rad both convert cleanly.
Formula
Angles convert through degrees: degrees = value × from-unit factor, then target = degrees ÷ to-unit factor. Exact factors: 1 rad = 57.29577951308232° · 1 gon = 0.9° · 1 arcmin = 1/60° · 1 arcsec = 1/3600° · 1 turn = 360°. Degrees to radians is × π/180, and radians to degrees is × 180/π.
Example
A right angle is 90° = 90 × π/180 = 1.5707963268 rad, which is why 90° and π/2 sit on the same line of a trig table. A 45° slope is 0.7853981634 rad and exactly 50 gon. Surveyors cut a degree into 60 arcminutes, so 0.5° is 30 arcmin and one degree spans 3,600 arcseconds. A full turn is 360° or 6.283185307 rad, and setting a calculator to radians makes sin(90) return 0.894 rather than 1.
Things to remember
Degrees survive because they are intuitive; radians survive because they are natural. Keep both on one screen, quote small angles in arcminutes where precision matters, and remember that 2π rad is one turn — the identity that keeps every rotation formula consistent.
Frequently Asked Questions
How many radians are in 90 degrees?
π/2 radians, which is 1.5707963268 to ten decimals. Multiply degrees by π/180 for the general case: 90 × 3.141592653589793 ÷ 180 = 1.5707963268. That is why a quarter circle in calculus is written π/2 and not 90.
What is a gradian and where is it used?
A gradian divides a right angle into 100 parts, so a full turn is 400 gon and 1 gon = 0.9°. Surveying equipment, theodolites and European civil-engineering drawings use it because decimal subdivision avoids minutes and seconds entirely.
Why does my calculator give the wrong trigonometric answer?
It is probably in the wrong angle mode. With 90 typed in radians, sin(90) = 0.894; switched to degrees the same keypress returns 1. Check the RAD or DEG indicator before trusting any sine, cosine or tangent, and convert the angle here first if the source uses another unit.
How do I convert arcminutes or arcseconds to degrees?
Divide by 60 for arcminutes and by 3,600 for arcseconds, because one degree is cut into 60 arcminutes and each arcminute into 60 arcseconds. A bearing of 51° 30′ is 51.5°, and a star position of −1° 22′ 30″ is −1.375° in decimal degrees.
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