How to use the degrees and radians converter
This converter moves an angle between degrees and radians. Rather than only giving a decimal, it also prints the exact multiple of π, adds gradians and turns on the same screen, and includes a third box where you can enter a multiple of π directly and get the angle back.
The formulas are radians = degrees × π ÷ 180 and degrees = radians × 180 ÷ π. Radians are irrational and cannot be written exactly as a finite decimal, so they are shown to 10 decimal places while degrees are shown to 6. The value of π in use and the degree value of 1 rad are printed on the constants line.
The exact π multiple comes from writing the degree value as a fraction over 180 and reducing it. 90° becomes π/2, 22.5° becomes π/8 and 270° becomes 3π/2, while 0° is simply 0. Decimal degrees convert exactly as long as they have at most six decimal places, which is why input is limited that way.
When you enter radians, the decimal multiple is shown and a search over denominators up to 64 looks for a simple π fraction that matches within a tolerance of 1e-9; if one exists it appears on an extra line. Entering 1.5707963268 yields 90° together with π/2, while a value with no close simple fraction leaves that line hidden.
Frequently Asked Questions
Radians = degrees × π ÷ 180, and degrees = radians × 180 ÷ π. Both come from the fact that 180° is exactly π radians.
The degree value is written as a fraction over 180 and reduced by the greatest common divisor. 90° is 90/180 = 1/2, giving π/2, and 22.5° is 45/360 = 1/8, giving π/8.
A gradian splits a right angle into 100 parts, so 1 gon is 0.9°, while a turn is one full revolution, so 360° is 1 turn. Both are handy for surveying and rotation work.